Holonomy invariants of links and nonabelian Reidemeister torsion
HHOLONOMY INVARIANTS OF LINKS AND NONABELIANREIDEMEISTER TORSION
CALVIN MCPHAIL-SNYDER
Abstract.
We show that the reduced SL ( C )-twisted Burau representationcan be obtained from the quantum group U q ( sl ) for q = i a fourth root ofunity and that representations of U q ( sl ) satisfy a type of Schur-Weyl dualitywith the Burau representation. As a consequence, the SL ( C )-twisted Reide-meister torsion of links can be obtained as a quantum invariant. Our construc-tion is closely related to the quantum holonomy invariant of Blanchet, Geer,Patureau-Mirand, and Reshetikhin [5], and we interpret their invariant as atwisted Conway potential. Contents
1. Introduction 21.1. Torsions of links 41.2. The Conway potential as a square root of the torsion 41.3. Colored braids and holonomy ribbon categories 41.4. Summary of the constructions 51.5. Overview of the paper 7Acknowledgements 72. The colored braid groupoid 82.1. Factorized groups 92.2. Enhanced colorings 133. Torsions and the Burau representation 143.1. Twisted homology and cohomology 153.2. Twisted Burau representations 163.3. Torsions 184. The algebra U i ( sl ) 194.1. Quantum sl at a fourth root of unity 194.2. Weight modules for U U i ( sl ) 265.1. The category C of weight modules 275.2. The opposite category of C C a r X i v : . [ m a t h . QA ] M a y CALVIN MCPHAIL-SNYDER F and F D U ⊗ n D U Introduction
Let X be a space and G a Lie group. We can capture geometric informationabout X by equipping it with a representation ρ : π ( X ) → G , considered up toconjugation. In this paper we consider the case of X = S \ L a link complementand G = SL ( C ). We call the pair ( L, ρ ) of the link L and representation ρ : π L → SL ( C ) a SL ( C ) -link, where π L := π ( S \ L ) is the fundamental group of thecomplement.To extend the representation of links as braid closures to this context, we usethe idea of a colored braid . Express the link L as the closure of a braid β on n strands. Topologically, we can think of β as an element of the mapping class groupof an n -punctured disc D n . Because π ( D n ) is a free group, we can equip the disc D n with a representation ρ : π ( D n ) → SL ( C ) by picking colors g i ∈ SL ( C ), with g i giving the holonomy of a path going around the i th puncture.The braid β acts on the colors by mapping ρ to the representation ρβ − . If L isthe closure of β , the representation ρ extends to a representation of the complementof L exactly when ρ = ρβ − . This perspective is one way to obtain invariants of G -links. The braid group (as the mapping class group of D n ) acts on the ρ -twistedcohomology of D n . In particular, its action on H ( D n ; ρ ) is the (twisted) Buraurepresentation, which can be used to define the twisted Reidemeister torsion of( L, ρ ).In this paper, we connect this story to the representation theory of the quantumgroup U = U q ( sl ) at q = i a fourth root of unity. Previous work [15, 16, 25] hasshown that the variety of SL ( C )-representations of D n is birationally equivalent toa central subalgebra of U ⊗ n and that the braid action on the representation varietycorresponds to the braiding on the quantum group.We extend this result to Burau representations. The reduced Burau representa-tion acts on a certain subspace H (cid:48) ⊂ H ( D n ; ρ ) of the ρ -twisted cohomology of the n -punctured disc. We find a subspace H (cid:48) n ⊂ U ⊗ n corresponding to H (cid:48) such thatthe braid action on H (cid:48) n gives the Burau representation. In addition, H (cid:48) n generatesa Clifford algebra C (cid:48) n inside of U ⊗ n . C (cid:48) n and U satisfy a sort of Schur-Weyl duality: C (cid:48) n is contained in the super-commutant of the action of U (via the coproduct) on This data is equivalently described by a G -local system on X or a gauge class of flat g -connections. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 3 U ⊗ n . This extends a similar result for U q ( gl (1 | ( C )-twisted torsion of alink can be obtained as a quantum invariant. Specifically, we define an invariant ∇ ( L, ρ, ω )of SL ( C )-links along with some extra data ω (a choice of square root for eachcomponent of L .) Strictly speaking, our construction also requires an orientationof L : see Remark 2.3.This invariant is essentially the quantum invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin [5] for q a fourth root of unity, differing only in a choiceof normalizaton (see Proposition 6.2.) We show that the norm-square of ∇ is thetorsion, in the sense that ∇ ( L, ρ, ω ) ∇ ( L, ρ, ω ) = τ ( L, ρ )where τ ( L, ρ ) is the ρ -twisted torsion of the complement of L , and ( L, ρ ) is themirror image of L . We therefore interpret ∇ as a nonabelian Conway potential. We also show that ∇ ( L, ρ, ω ) is gauge-invariant. That is, for any g ∈ SL ( C ),we can consider the conjugated representation ρ g ( x ) = gρ ( x ) g − , and we show that ∇ ( L, ρ, ω ) = ∇ ( L, ρ g , ω )A meaningful invariant of G -links should be gauge-invariant, as changing the base-point of π ( S \ L ) conjugates the representation ρ . (Similarly, one could changebasis in the space C on which SL ( C ) acts.) The torsion is known to be gauge-invariant.This terminology comes from the case where G is a Lie group with Lie algebra g . In that case the representation ρ can be described as the holonomy of a flat g -valued connection on the complement of L , and global conjugation correspondsto a gauge transformation of the connection.The reader may recall that quantum sl -invariants are closely related to thephysics concept of a Chern-Simons topological quantum field theory with gaugegroup SU(2). Our construction is a variant of this: we define a Chern-Simons theorywith gauge group SL ( C ) for links in S . The sl -connection corresponding to ρ issomething like a background magnetic field. “Physical quantities” (mathematicallyinteresting invariants) should depend only on the gauge class of the backgroundfield.Our invariant ∇ ( L, ρ, ω ) has two technical requirements: that the colored link L admit a presentation as the closure of an admissible braid (see §
2) and that tr ρ ( x ) (cid:54) =2 for every meridian x of L . The first, which is related to the fact that SL ( C ) ∗ is only birationally equivalent to SL ( C ), is not particularly important, becauseevery ( L, ρ ) is gauge-equivalent to one with an admissible braid presentation. Thesecond requirement is also to be expected, because the torsion can be ill-definedwhen det(1 − ρ ( x )) = 2 − tr( ρ ( x )) = 0 for meridians x of L , since then the cochaincomplex H ∗ ( S \ L ; ρ ) can fail to be acyclic.The results of this paper are mostly algebraic, not topological: we show how toreproduce a known invariant, the torsion, in terms of quantum groups. However, Typically the Conway potential is viewed as being a symmetrized version of the Alexanderpolynomial, not its square root. The difference has to do with using SL -torsions instead ofGL -torsions, and is explained in more detail in § CALVIN MCPHAIL-SNYDER we hope that future work in this direction could relate geometric invariants like thetorsion with quantum invariants like the colored Jones polynomial, with potentiallysignificant topological consequences.1.1.
Torsions of links.
The untwisted Reidemeister torsion τ ( L ) of a link com-plement S \ L (which is essentially the Alexander polynomial of L ) is defined usingthe representation ρ : π L → GL ( Q ( t )) sending each meridian x of π L to t . Moregenerally one can send all meridians in component i to a variable t i , which givesthe multivariate Alexander polynomial.The torsion is defined using the ρ -twisted (co)homology of S \ L and still makessense for ρ a representation into any matrix group GL n ( k ) for k a field, as longas the ρ -twisted homology is acyclic. When the image of ρ is nonabelian, τ ( L, ρ )is usually called the twisted torsion. We prefer to call the two cases abelian and nonabelian torsion, since a twisted chain complex occurs in both. Recently therehas been considerable interest in nonabelian torsions of links; one overview is [9].It is known [20, 21] that the abelian torsion can also be obtained from thequantum group U q ( gl (1 | U i ( sl ). Our workextends this construction to the case of nonabelian SL ( C ) torsions.1.2. The Conway potential as a square root of the torsion.
We explain theinterpretation of ∇ as a nonabelian Conway potential. The classical Reidemeistertorsion τ ( L ) is only defined up to an overall power of t . It is possible to refinethe torsion to a rational function ∇ ( L, √ t ) of √ t , the Conway potential, which isdefined up to an overall sign. In fact, ∇ is always of the form ( √ t − (cid:112) /t ) − p ( t ) forsome symmetric Laurent polynomial p , so we can think of p ( t ) as a symmetrizedAlexander polynomial.One way to construct the Conway potential is as follows: Instead of sending eachmeridian to t , consider the representation α into SL ( Q ( t )) sending the meridiansto (cid:18) t t − (cid:19) Then the Reidemeister torsion τ ( L, α ) is defined up to ± det α = ±
1. Furthermore,it always factors as a product τ ( L, α ) = ∇ ( L, √ t ) ∇ ( L, −√ t )Our invariant ∇ ( L, ρ, ω ) is analogous for the nonabelian case, with the choice ofsquare roots ω generalizing the choice of square root √ t .Another perspective (see [22, § oriented link L the sign of ∇ ( L, √ t )is fixed, unlike the sign of τ ( L, α ). Our extension ∇ ( L, ρ, ω ) does not satisfy thisproperty, since even with an orientation of L it is only defined up to a fourth rootof unity.1.3. Colored braids and holonomy ribbon categories.
The present work fo-cuses on a holonomy, or G -graded, version of the Reshetikhin-Turaev construction Picking an orientation of L gives an orientation on its meridians, so one can distinguishbetween t and t − , hence fix the sign of √ t − (cid:112) /t . See also Remark 2.3. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 5 for braids. We recall the original construction briefly. If H is a ribbon Hopf al-gebra and V a finite-dimensional H -module, the quasitriangular structure (i.e. the R -matrix) of H yields a representation F : B n → End H ( V ⊗ n )of the braid group on n strands (viewed as a category with one object) in thecategory of H -modules. The ribbon structure on H gives a quantum trace tr q generalizing the usual trace, and the scalartr q F ( β )is an invariant of the (framed) link L obtained as the closure of β . This con-struction gives quantum invariants of links, such as the Jones polynomial and itsgeneralizations.To upgrade to the G -graded case, we think of the braid group as the map-ping class group of a punctured disc and equip the disc with a representation ρ : π ( D n ) → G . To keep track of the different representations, we replace the braidgroup with a colored braid groupoid . Objects of this groupoid can be algebraicallydescribed as tuples g = ( g , . . . , g n ) of elements of G . Morphisms β : g → g (cid:48) arebraids on n strands sending g to g (cid:48) . We explain the braid ation on the colors inmore detail in Section 2.For now, we can see that a Reshetikhin-Turaev-type colored braid groupoidrepresentation should send β to an intertwiner of H -representations V g ⊗ · · · ⊗ V g n → V g (cid:48) ⊗ · · · ⊗ V g (cid:48) n where V − is now a family of H -modules parametrized by G . The closure of β has awell-defined G -representation if and only if β is an endomorphism of this groupoid,and taking traces gives quantum holonomy invariants of links. We will call a functorof this type a quantum holonomy representation of colored braids, generalizing thequantum representations of ordinary braids.We briefly describe how to obtain such G -graded modules. At q = ξ a rootof unity, the quantum group U ξ ( sl ) contains a large central subalgebra Z , soits representations are parametrized by points of Spec Z . The Hopf structure on U ξ ( sl ) makes Z into an algebraic group, in this case SL ( C ) ∗ , the Poisson dual ofSL ( C ). These two groups are different, but they are birationally equivalent, andthis is enough to obtain the desired grading on modules. This difference explainsthe use of the factorized biquandle associated to SL ( C ) instead of the simplerconjugation quandle of SL ( C ). These concepts are explained in more detail inSection 2.1.4. Summary of the constructions.
We summarize the invariants, includingthe topological interpretations to be shown later. Let (
L, ρ, ω ) be an enhancedSL ( C )-link; here enhanced refers to the choice of square roots ω . We define func-tors:(1) F : ˆ B (SL ( C )) ∗ → C , which defines an invariant ∇ ( L, ρ, ω ).(2) F : ˆ B (SL ( C )) ∗ → C , corresponding to an invariant ∇ ( L, ρ, ω ). It is nothard to see (see Proposition 6.10) that ∇ ( L, ρ, ω ) = ∇ ( L, ρ, ω )where ( L, ρ ) is the mirror image of (
L, ρ ). CALVIN MCPHAIL-SNYDER (3) T : ˆ B (SL ( C )) ∗ → D , which defines an invariant T ( L, ρ, ω ). It is immediatefrom the defintion of T that T ( L, ρ, ω ) = ∇ ( L, ρ, ω ) ∇ ( L, ρ, ω ) . Our main result (Theorem 8) is that T ( L, ρ, ω ) = τ ( L, ρ )is the ρ -twisted torsion of the complement of L . The proof uses the Schur-Weyl duality of § B (SL ( C )) ∗ is an extension (related to the choices of square roots) of the braidgroupoid B (SL ( C )) ∗ associated to the factorized group SL ( C ) ∗ . The categoriesappearing in the images of these functors are(1) C , the category of nonsingular weight modules for the quantum group U i ( sl ),(2) C , a graded opposite of the braided category C , and(3) D , a certain subcategory of the Deligne tensor product C (cid:2) C . We can thinkof D as the homogeneous part of C (cid:2) C .We define D as a subcategory of the category of weight modules for the algebra U i ( sl ) (cid:2) U i ( sl ) = U i ( sl ) ⊗ C U i ( sl ), so we do not really need the Deligne productin full generality. However, we use the notation (cid:2) in § § U ⊗ ni (cid:2) U ⊗ ni ; these arise when studying thedecomposition of tensor products of ( U i ⊗ C U i )-modules.For the reader familiar with algebraic TQFT, the following discussion may helpmotivate these constructions. The invariant of (enhanced) SL ( C )-links in S con-structed from C is a surgery or Reshetikhin-Turaev invariant of link complements.This theory is anomalous because the representations involved are projective. However, in the doubled theory D , the anomalies from C and C cancel.One could think of the invariant from D as being the state-sum or Turaev-Viro invariant associated to C . For the non-graded case, it is well-known that the state-sum theory on a fusion category C agrees with the surgery theory on the Drinfeldcenter Z ( C ). For more details, see the book [23] by Turaev and the series of papers[3, 1, 2] by Balsam and Kirilov Jr. If C is modular (in particular, if it has a braiding)then there is an equivalence of categories C (cid:2) C ≡ Z ( C ) [19], so we can compute thevalue of the state-sum theory from C by using the surgery theory from C (cid:2) C .In the G -graded case, Turaev and Virelizier [24] define notions of state-sum andsurgery homotopy quantum field theory (a.k.a. G -graded TQFT) and show that thestate-sum theory from C is equivalent to the surgery theory from Z G ( C ), where Z G ( C ) is a graded version of the Drinfeld center of C . We conjecture that (as inthe non-graded case) there should be an equivalence Z G ( C ) ∼ = D so that we can interpret our surgery invariant from D as being the state-sum versionof the invariant of C . Usually, the theory for link complements is not anomalous; the anomaly instead appears forgeneral manifolds resulting from surgery. In the holonomy case, the anomalies show up earlier,because U i is no longer quasitriangular. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 7
In the non-graded case, it is well-known that the Drinfeld center corresponds tothe Drinfeld double, in the sense that there is an equivalence of braided categories Z (Rep( H )) ∼ = Rep( D ( H ))where H is a (not necessarily quasitriangular) Hopf algebra. For this reason, weconjecture that there is a graded version D G of the Drinfeld double constructionsuch that D ∼ = Rep( D G ( U i ( sl ))) . It is likely that this construction is related to the work of Zunino [26, 27] on crossedquantum doubles.1.5.
Overview of the paper. • Section 2 fixes conventions on colored braids and discuss the factorizationstructure used to relate SL ( C ) and SL ( C ) ∗ -colorings. • Section 3 defines the twisted Burau representations and the twisted Reide-meister torsion. • Section 4 defines the algebra U = U i ( sl ) and gives some results on itsrepresentation theory and quasitriangular structure. • Section 5 expands on this theory by defining the categories C , C , and D constructed from representations of U , and discusses the modified traces onthese categories. • Section 6 uses the results of Sections 4 and 5 to define colored braid groupoidrepresenations and the associated link invariants and proves some basicproperties of these invariants. • Section 7 discusses the Schur-Weyl duality between U and the twisted Buraurepresentation and applies it to prove the main result: that T ( L, ρ ) agreeswith the torsion. • Appendix A contains more details of the construction of the modified tracesintroduced in Section 5.
Acknowledgements
I would like to thank Nicolai Reshetikhin for introducing me to the problem andfor many helpful suggestions, Noah Snyder for an enlightening conversation thatlead me to the correct definition of the doubled representation T , and ChristianBlanchet, Hoel Queffelec, and N.R. for sharing some unpublished notes [4] on ho-lonomy R -matrices. In addition, I want to thank Bertrand Patrueau-Mirand forfinding the right argument for (3) of Proposition 3.6, as well as Nathan Geer forhis talk explaining the modified trace construction in the appendix.While none of the computations in this paper require computer verification, thecomputer algebra system SageMath and programming language Julia were veryhelpful in intermediate work, and I thank the developers and maintainers of thissoftware for their work.During the final preparation of this article, I was saddened to learn of the passingof John H. Conway. He made remarkable contributions to many areas of mathemat-ics, and I particularly admire his work in knot theory on link potential functionsand algebraic tangles. At the conference
New Developments in Quantum Topology at UC Berkeley in June 2019.
CALVIN MCPHAIL-SNYDER
Figure 1.
The braid σ σ − σ . x x x x x = x x x x = x x x x = x x Figure 2.
Wirtinger generators of the trefoil group and the rela-tion at each crossing.2.
The colored braid groupoid
Conventions 2.1.
The braid group B n on n strands has generators σ , . . . , σ n − ,with σ i given by braiding strand i over strand i +1. Braids are drawn and composedleft-to-right. For example, Figure 1 depicts the braid σ σ − σ on 3 strands.Let L be a link in S . Given a diagram of L , we obtain the Wirtinger presentation of the group π L = π ( S \ L ). (See Figure 2.) This presentation assigns onegenerator x i to each arc (unbroken curve) and one conjugation relation to eachcrossing. If we represent L as the closure of a braid β on n strands, we can examinethe interaction between this presentation and the braid group.Specifically, the Wirtinger presentation gives an action of the braid group B n onthe free group F n . We can think of putting free generators x , . . . x n ∈ F n on the n strands on the left and acting on them by the braid to get words on the right.Concretely, the generators act by(1) x j · σ i = x − i x i +1 x i j = i,x i j = i + 1 ,x j otherwise.as in Figure 3. Here the braid action on the free group F n is written on the right,to match left-to-right composition of braids.It follows that for any braid β with closure L , π L = (cid:104) x , . . . , x n | x i = x i · β (cid:105) gives a presentation of the fundamental group of S \ L . In particular, a choice ofrepresentation ρ : π L → G of the complement of the closure L is equivalent to achoice of group elements ρ ( x i ) such that ρ ( x i ) = ρ ( x i · β ) for each i . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 9 x x x x − x x Figure 3.
Braid action on the free group.
Definition 2.2. A G -colored braid is a braid β on n strands and a tuple ( g , . . . , g n )of elements of G . The G -colored braid groupoid is the category B ( G ) whose objectsare tuples ( g , . . . , g n ) and whose morphisms are braids( g , . . . , g n ) β −→ ( ρ ( x · β ) , . . . , ρ ( x n · β ))where ρ : F n → G is defined by ρ ( x i ) = g i . In particular, braid generators act by( g , g ) σ −→ ( g − g g , g )One can think of the union B := (cid:83) n B n of the braid groups as a category withobjects { , , . . . } , and links can be represented as closures of endomorphisms of B .Similarly, links with a representation ρ : π L → G can be represented as closuresof endomorphisms of B ( G ). B ( G ) is a monoidal category in the usual way: theproduct of objects is their concatenation, and the product of morphisms is obtainedby placing them in parallel. In our conventions, this monoidal product is verticalcomposition. Remark 2.3.
The presentation of a G -link ( L, ρ ) as the closure of a braid β ∈ B ( G )implicitly requires a choice of orientation. There are distinguished meridians x i around the base of the braid, but choosing between ρ ( x i ) = g i and ρ ( x i ) = g − i requires an orientation of the meridian x i .The usual way to do this is to orient L and use this to obtain an orientationof the meridian. For example, consider the result that the Conway potential is asign-refined version of the Alexander polynomial defined for oriented links.We will usually leave this choice implicit going forward, but it will come up againwhen we discuss the mirrored invariants ∇ in Proposition 6.10.2.1. Factorized groups.
To deal with the fact that the central subalgebra Z of U i ( sl ) is not (the algebra of functions on) SL ( C ) but its Poisson dual groupSL ( C ) ∗ we need to use a slightly different description of SL ( C )-links. Definition 2.4. A generalized group factorization is a triple ( G, G, G ∗ ) of groups,with G a normal subgroup of G , along with maps ϕ + , ϕ − : G ∗ → G such that themap ψ : G ∗ → G ψ ( a ) = ϕ + ( a ) ϕ − ( a ) − restricts to a bijecton G ∗ → G .The example to keep in mind is More generally, one could define the groupoid of braids colored by any quandle. Here werestrict to conjugation quandles of groups.
Example 2.5.
The
Poisson dual group of SL ( C ) isSL ( C ) ∗ := (cid:26)(cid:18)(cid:18) κ ϕ (cid:19) , (cid:18) ε κ (cid:19)(cid:19)(cid:27) ⊆ GL ( C ) × GL ( C )Set G = SL ( C ) and G = GL ( C ) × GL ( C ), and let ϕ + , ϕ − : G ∗ → G be theinclusions of the first and second factors. Then the map ψ acts by(2) ψ : (cid:18)(cid:18) κ ϕ (cid:19) , (cid:18) ε κ (cid:19)(cid:19) (cid:55)→ (cid:18) κ − εϕ (1 − εϕ ) /κ (cid:19) This is not a generalized group factorization, because not all matrices in SL ( C )occur on the right-hand side of (2).The map ψ is a bijection between SL ( C ) ∗ and the Zariski open subset U ofSL ( C ) of matrices with upper-left entry nonzero, so we call this a generic groupfactorization. We will show that every link admits a presentation with holonomieslying in U so that we can use SL ( C ) ∗ colorings instead of SL ( C ) colorings andthus use the braiding on U i ( sl ).We first describe how to use the group factorization to associate a tuple( g , . . . , g n ) ∈ SL ( C ) × · · · × SL ( C )of SL ( C ) elements to a tuple( a , . . . , a n ) ∈ SL ( C ) ∗ × · · · × SL ( C ) ∗ of SL ( C ) ∗ elements, in a way respecting the braiding action on the colors. This isa special case of the biquandle factorization defined in [5].Write a ± for ϕ ± ( a ). Then ψ extends to a map on tuples ψ = ( ψ , · · · , ψ n ) with(3) ψ i ( a , . . . , a n ) = ( a +1 · · · a + i − ) ψ ( a i )( a +1 · · · a + i − ) − where ψ ( a i ) = a + i ( a − i ) − . The formula is somewhat nicer in terms of the products g i · · · g : ( ψ i · · · ψ )( a , . . . , a n ) = a +1 · · · a + i ( a − · · · a − i ) − , i = 1 , . . . , n This is best-understood graphically. For example, the blue path in Figure 4corresponds to the image g of the generator x of the fundamental group of thetwice-punctured disc. As it crosses the dashed line above the first point from leftto right, it picks up a factor of a +1 , then a +2 for the next dashed line. When crossingthe line below, we get a factor for ( a − ) − because we are crossing right to left, andsimilarly for ( a +1 ) − . We have derived the relation g = ψ ( a , a ) = a +1 a +2 ( a − ) − ( a +1 ) − . We can think of the a i as local coordinates and the g i as global coordinates. Asan explicit example, if a i = (cid:18)(cid:18) κ i ϕ i (cid:19) , (cid:18) ε i κ i (cid:19)(cid:19) for i = 1 ,
2, then the expressions for the images g = ψ ( a , a ) = a +1 ( a − ) − g = ψ ( a , a ) = a +1 a +2 ( a − ) − ( a +1 ) − SL ( C ) is a Poisson-Lie group, so its Lie algebra sl is a Poisson-Lie bialgebra. There is adual Poisson-Lie bialgebra sl ∗ , and its associated Lie group is SL ( C ) ∗ . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 11 a +1 a +2 a − a − g g g Figure 4.
Derivation of g = a +1 a +2 ( a − ) − ( a +1 ) − (blue path) and g g = a +1 a +2 ( a − a − ) − (black path.) a a a a Figure 5.
Derivation of the biquandle relation a − a +2 = a +4 a − in (4).of the Wirtinger generators are somewhat complicated, while the expressions fortheir products g = a +1 ( a − ) − = (cid:18) κ − ε ϕ − ε ϕ κ (cid:19) g g = a +1 a +2 ( a − a − ) − = (cid:18) κ κ − ε κ − ε κ ϕ + ϕ − ( ε κ + ε )( κ ϕ + ϕ ) κ κ (cid:19) are simpler.If σ : ( a , a ) → ( a , a ) is a generator, the image colors are the unique solutionsto the equations(4) a +1 a +2 = a +4 a +3 , a − a − = a − a − , a − a +2 = a +4 a − which we can read off by thinking about paths above, below, and between thestrands. For example, a − a +2 = a +4 a − follows from comparing the red (left) andblue (right) paths in Figure 5. Definition 2.6.
Let a , a ∈ SL ( C ) ∗ . When they exist, let a , a be the uniquesolutions of (4) and set B ( a , a ) = ( a , a ). We say that B and SL ( C ) ∗ form a generic biquandle .For a general definition, see [5, § generic biquandle. The generaltheory of this is dealt with in [5, § B is defined. Definition 2.7. B (SL ( C )) ∗ is the category whose objects are tuples ( a , . . . , a n )of elements of SL ( C ) ∗ and whose morphisms are admissible colored braids be-tween them, with the action on colors given by the map B . A braid generator σ : ( a , a ) → B ( a , a ) is admissible if B ( a , a ) is defined (i.e. if the equations(4) have a solution) and a colored braid is admissible if it can be expressed as aproduct of admissible generators.We refer to morphisms of B (SL ( C )) ∗ as SL ( C ) ∗ -colored braids . B (SL ( C )) ∗ becomes a monoidal category in the usual way, with the product of objects givenby concatenation and the product of braids given by vertical stacking.Notice that the statement that β is a SL ( C ) ∗ -colored braid includes the claimthat it is admissible. An admissible SL ( C )-colored braid is one that is equivalentto an admissible SL ( C ) ∗ braid. Proposition 2.8.
The map (3) gives a functor B (SL ( C )) ∗ → B (SL ( C )) .Proof. This is a special case of [5, Theorem 3.9]. (cid:3)
This functor is not an equivalence of categories because it is not onto, but itcan be shown to be a generic equivalence, in a sense made precise in [5, § Definition 2.9.
Let ( g , . . . , g n ) be an object of the SL ( C )-colored braid groupoid,that is a tuple of elements of SL ( C ). We say it is admissible if for i = 1 , . . . , n theelement g i · · · g has a nonzero 1 , σ i : ( g , · · · , g n ) → ( g , · · · , g n ) is admissible if its source and target areadmissible, and a SL ( C )-colored braid β is admissible if it can be expressed as aproduct of admissible generators. Proposition 2.10.
Every SL ( C ) -link L is gauge-equivalent to one admitting apresentation as the closure of an admissible SL ( C ) -colored braid β , hence as theclosure of a SL ( C ) ∗ -colored braid.Proof. L is clearly the closure of some SL ( C )-colored braid β : ( g , . . . , g n ) → ( g , . . . , g n ). (It is closure of a braid β , and the representation ρ makes β a coloredbraid.) If ( g , . . . , g n ) is not admissible, we can conjugate ρ to obtain an admissibleobject. Now by [5, Theorem 5.5] β can be written as an admissible product ofgenerators, hence is admissible. (cid:3) It follows that we can obtain invariants of SL ( C )-links by the following process:(1) Construct Markov traces (see Definition 6.5) of SL ( C ) ∗ -braids. These area colored generalization of the usual notion of a Markov trace on the braidgroup. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 13 (2) Gauge transform the SL ( C )-link ( L, ρ ) to a link (
L, ρ g ) that is the closureof an admissible SL ( C )-braid β .(3) Because β is admissible, it can be pulled back to a SL ( C ) ∗ -braid β , andthe Markov trace t of this braid will be an invariant of ( L, ρ g ).(4) If the Markov trace is gauge-invariant in an appropriate sense, then t willalso be an invariant of ( L, ρ ).Later, to get the relationship with the torsion, we will need one more condition,which is that the SL ( C )-braid β has nontrivial total holonomy. For links withwell-defined torsion this is always the case; see Proposition 3.6.We conclude with a computation that will be useful later. Lemma 2.11.
Suppose B ( a , a ) exists and is equal to ( a , a ) . Then there is acomplex number κ (cid:54) = 0 with a = a = (cid:18)(cid:18) κ
00 1 (cid:19) , (cid:18) κ (cid:19)(cid:19) equivalently ψ ( a ) = ψ ( a ) = (cid:18) κ κ − (cid:19) Proof.
This follows immediately from writing the equations (4) in coordinates. (cid:3)
Enhanced colorings.
For technical reasons, our invariant depends not onlyon the SL ( C )-link ( L, ρ ) but also a choice of square root of tr( ρ ( x i )) − x i ; this choice specifies the action of the quantum Casimir element Ω ∈ U . Definition 2.12.
Let (
L, ρ ) be a link with components L , . . . , L k . Choose merid-ians x i of each component L i . A system of Casimirs ω = ( ω , . . . , ω k ) for ( L, ρ ) isa choice of complex numbers ω i such that ω i = tr( ρ ( x i )) − i = 1 , . . . , k . We call the triple ( L, ρ, ω ) an enhanced SL ( C ) -link .It is not hard to see from the Writinger presentation that the meridians of onecomponent are all conjugate. Since the trace is conjugation-invariant, it followsthat the number tr( ρ ( x i )) − ρ ( x i ) has eigenvalues λ, λ − , thentr( ρ ( x i )) − λ − λ − = ( λ / − λ − / ) so this choice is closely related to taking a square root of the eigenvalues of ρ ( x i ).If we represent L as the closure of a SL ( C ) ∗ -colored braid β : ( a , . . . , a n ) → ( a , . . . , a n ), then the images of the meridians are a +1 · · · a + i − a + i ( a − i ) − ( a +1 · · · a + i − ) − which are conjugate to the elements ψ ( a i ) = a + i ( a − i ) − so we can equivalently give square roots of tr( ψ ( a i )) − σ : ( a , a ) → ( a , a ) be an SL ( C ) ∗ -colored braid generator. We can checkthat a , a and a , a lie in the same conjugacy class, so the braid group acts onour choice of Casimir by permutations. Definition 2.13.
The enhanced SL ( C ) ∗ -colored braid groupoid is a categoryˆ B ∗ (SL ( C ))extending B ∗ (SL ( C )). Objects are tuples(( a , ω ) , · · · , ( a n , ω n ))with a i ∈ SL ( C ) ∗ and ω i = tr( ψ ( a i )) −
2. Braids act on the a i by the SL ( C ) ∗ biquandle action and on the ω i by permutations. Explicitly, if ( a , a ) = B ( a , a ),then B (( a , ω ) , ( a , ω )) = (( a , ω ) , ( a , ω ))By extending the earlier description of SL ( C )-links as closures of SL ( C ) ∗ -colored braids, it is now clear that enhanced SL ( C )-links can be described asclosures of endomorphisms of ˆ B (SL ( C )) ∗ .One way to understand this construction is as follows. The generic biquandle X underlying ˆ B ∗ (SL ( C )) is a fiber product of the generic biquandle SL ( C ) ∗ and theset C via the diagram X SL ( C ) ∗ C C fg where f ( a ) = tr( ψ ( a )) − g ( z ) = z . (See [5, Example 3.16, and also p. 31].)We will see in § U = U i ( sl ) corre-sponds to the groupoid ˆ B ∗ (SL ( C )).3. Torsions and the Burau representation
We define the torsion via the twisted Burau representation of the colored braidgroupoid, which is obtained via the ρ -twisted cohomology of the punctured disc.Many of our ideas are essentially the same as those of A. Conway [6]. Most authorsuse homology instead of cohomology, but these give equivalent torsions becausePoincar´e duality holds for the local systems we are considering.To describe representations π ( S \ L ) → SL ( C ), we represent L as a braidclosure of a braid β on n strands. By doing this, we place L inside a solid torus T .We can slice T open across a meridional disc D n , which we think of as having n punctures corresponding to the strands of β .From this perspective, we can view the algebraic category B ( G ) of the previoussection as a model for a topological category D ( G ). This category has objects pairs( D n , ρ ), where D n is an n -punctured disc and ρ is a representation π ( D n ) → G .The morphisms are f : ( D n , f ∗ ρ ) → ( D n , ρ )for f an element of the mapping class group of D n , where f ∗ ρ = ρ ◦ f is thepullback. As π ( D n ) = F n is a free group, representations π ( D n ) → G are n -tuples of elements of G , and since the mapping class group of D n is B n , it is nothard to see that D ( G ) is equivalent to B ( G ).The point of this topological description is that we obtain a colored braid actionon the twisted cohomology of D n . We recall the definition below. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 15
Twisted homology and cohomology.
Let X be a finite CW complex withfundamental group π = π ( X ), and let ρ : π → GL( V ) be a representation, where V is a vector space over C . We think of this as a right representation acting onrow vectors, so that V is a right Z [ π ]-module.Let ˜ X be the universal cover of X , and pick lifts ˜ e ki of the cells of X . (We write e k , e k , . . . for the cells of dimension k .) π = π ( X ) acts on the cells of the universalcover, and this action commutes with the differentials. We take this to be a leftaction, so that the cellular chain complex C ∗ ( ˜ X ) of the universal cover becomes acomplex of left Z [ π ]-modules. Definition 3.1.
The ρ -twisted homology H ∗ ( X ; ρ ) of X is the homology of the ρ -twisted chain complex C ∗ ( X ; ρ ) := V ⊗ Z [ π ] C ∗ ( ˜ X )We have given this definition in terms of a CW complex for X and a choice oflifts, but it can be shown to not depend on the choice of lifts. In fact, the ρ -twistedhomology also does not depend on the CW structure. One way to see this is togive a definition in terms of GL( V )-local systems.To match the braid action on multiplicity spaces in (16) we need to take duals.Consider the cochain complex C ∗ ( ˜ X ) := Hom Z ( C ∗ ( ˜ X ) , Z )It becomes a left Z [ π ]-module via x · f = e (cid:55)→ f ( x − e )This definition makes the natural pairing between C ∗ ( ˜ X ) and C ∗ ( ˜ X ) nondegenerateand π -equivariant. To twist the cochains we need a right Z [ π ]-module. The dualspace V ∗ := Hom C ( V, C ) is a right π -module via x · f = v (cid:55)→ f ( vρ ( x − ))We write ( V ∨ , ρ ∨ ) for this representation. Note that there is a π -equivariant pairingbetween V ∨ and V . x · ( f ⊗ v ) (cid:55)→ f ( ρ ( x − ) ρ ( x ) v ) = f ( v ) Definition 3.2.
The ρ -twisted cohomology H ∗ ( X ; ρ ) of X is the cohomology of the ρ -twisted cochain complex C ∗ ( X ; ρ ) := Hom Z [ π ] ( C ∗ ( ˜ X ) , V ∨ )where V ∨ is the dual representation to ( V, ρ ) thought of as a right module.When C ∗ ( X ; ρ ) is acyclic, so is C ∗ ( X ; ρ ), and we can use the pairing to showthat they have the same torsion. More generally this works for a module over any commutative ring; this perspective is impor-tant when defining the twisted Alexander polynomial.
Twisted Burau representations.
From now on, we fix a basis of V ∼ = C k .Recalling the topological interpretation of morphisms of B (GL( C k )) as braids actingon the punctured disc, we obtain a representation B ( β ) : H ( D n ; β ∗ ρ ) → H ( D n ; ρ )called the twisted Burau representation. Write x i for the loop around the i th puncture, so that π ( D n ) is the free groupon the generators x , . . . , x n . D n is homotopy equivalent to a wedge of n circles,and we can view the choice of generators as a choice of 1-cells e i . Along withthe usual basis e , . . . , e k of C k , the image of the lifts ˜ e , ˜ e , . . . , ˜ e n gives a basis { e i ⊗ ˜ e j } for each H ( D n ; ρ ). With respect to these bases we can compute matricesof B ( β ), which will nk × nk .The representation B is reducible. To see this, observe that the product x · · · x n of all the generators is invariant under the braid group action (1). If we introducegenerators y i := x · · · x i then we see that the braid group acts on them via(5) y j · σ i = (cid:40) y i − y − i y i +1 , j = iy j , j (cid:54) = i Definition 3.3.
Let H ( D n ; ρ ) (cid:48) be the subspace of H ( D n ; ρ ) spanned by the(cochains dual to the) images of the generators y i = x · · · x i for 1 ≤ i < n . The reduced twisted Burau representation is the restriction of B to maps B ( β ) : H ( D n ; β ∗ ρ ) (cid:48) → H ( D n ; ρ ) (cid:48) We write B ( ρ ) for the space H ( D n ; ρ ) on which the Burau representation acts.Let σ i : ρ → ρ be a colored braid generator. Then the matrices of the reducedtwisted Burau representations with respect to these bases are given by (cid:104) B ∨ ( σ i ) (cid:105) = I ( i − k ⊕ I k ρ ∨ ( y i +1 y − i ) 00 − ρ ∨ ( y i +1 y − i ) 00 I k I k ⊕ I ( n − i − k (6)To agree with the left-to-right composition of braids we view these matrices asacting on row vectors from the right. Recall that the matrices of ρ ∨ are the inversetransposes of the matrices of ρ .The above representation is very close to the action (16) on the quantum groupmultiplicity space, but to get them to match we need to change basis. It is more common to define the Burau representation in terms of homology, but this formmatches the quantum groups construction.
OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 17
Proposition 3.4.
There exists a family of bases of the cohomology H ( D n , ρ ) (cid:48) suchthat the matrices of the reduced Burau representation B are (7) I i − ⊕ κ − i − ϕ i κ − i − κ − i ϕ i κ − i − ε i +1 − κ i +1 ε i + i κ i +1 ⊕ I n − i − The above matrix is exactly the image of (16) under the Z -characters corre-sponding to ρ . Proof.
Let ρ be an admissible representation into SL ( C ) such that ρ is also ad-missible, so that we can think of σ i as an SL ( C ) ∗ -colored braid ( a , . . . , a n ) → ( b , . . . , b n ). To avoid cumbersome notation, we temporarily write a ± i for the trans-poses of the usual factorization matrices. Then if p i ( a ) := a + i · · · a +1 , m i ( a ) := a − i · · · a − i we have ρ ∨ ( y i ) = p i ( b ) − m i ( b ) ρ ∨ ( y i ) = p i ( a ) − m i ( a )and in particular ρ ∨ ( y i +1 y − i ) = p i +1 ( b ) − b − i +1 p i ( b ) ρ ∨ ( y i y − i − ) = p i ( a ) − a − i p i − ( a )Because the spaces H ( D n ; ρ ) are parametrized by the representation ρ , choosinga basis really means choosing a family of bases depending on ρ . The matrices (6)are written with respect to the bases { I ⊗ y ∗ i } n − i =1 , where I is the standard basisof C thought of as a matrix, and y ∗ i is the cochain dual to the image of y i . If weinstead use the bases { I ⊗ y ∗ n − i } n − i =1 , we obtain the matrices M ( σ i ) = I i − ⊕ I k I k − ρ ∨ ( y i +1 y − i ) 00 ρ ∨ ( y i +1 y − i ) I k ⊕ I n − i − = I i − ⊕ I k I k − p i ( a ) − a − i p i − ( a ) 00 p i +1 ( b ) − b − i +1 p i ( b ) I k ⊕ I n − i − We want to change basis once again by replacing the standard bases I of V ∗ ∼ = C by the bases p − j . Under this change the matrix M ( σ i ) transforms to M ( σ i ) (cid:55)→ M (cid:48) ( σ i ) = (cid:2) p j ( a ) (cid:3) j M ( σ i ) (cid:2) p j ( b ) − (cid:3) j = I i − ⊕ I ( b + i ) − a − i ( b + i ) − b − i +1 I ⊕ I n − i − (8) where we have used the fact that p j ( a ) = p j ( b ) for all j (cid:54) = i . Writing b j = ( b + j , b − j ) = (cid:18)(cid:18) κ j ϕ j (cid:19) , (cid:18) ε j κ j (cid:19)(cid:19) for the (transposed) coordinates of the target of σ i , we can write (8) as I i − ⊕ κ − i − ϕ i κ − i − κ − i ϕ i κ − i − ε i +1 − κ i +1 ε i + i κ i +1 ⊕ I n − i − as claimed. (cid:3) Torsions.
When the complex C ∗ ( D n ; ρ ) is acyclic (each H ∗ ( D n ; ρ ) is triv-ial) we can still extract an invariant, the torsion . Details on the classical case ofuntwisted/abelian torsions are found in the book [22]. Twisted torsions and therelated twisted Alexander polynomial are discussed in the article [6] and thesis [7],as well as the survey article [9].We sketch the definition of the torsion. Acyclicity is equivalent to exactness ofthe sequence · · · C i ∂ i −→ C i − · · · in which case we get isomorphisms Ker ∂ i = Im ∂ i − . If we choose a basis of each C i ,we can use the above isomorphisms to change these bases. The alternating productof determinants of the basis-change matrices gives an invariant of the acyclic com-plex C ∗ . The definition for cochain complexes is analogous. In general this torsioncan depend on the choice of basis for each chain space, but for link complements itdoes not.Given a presentation of L as the closure of a braid β we get a presentation of π L = (cid:104) y , . . . , y n | y i = y i · β (cid:105) , which in turn gives a CW structure on S \ L ; the2-cells are obtained by the relations y i = y i · β . Link complements are aspherical,so we don’t need to add any higher-dimensional cells. Definition 3.5.
Let ρ : π ( S \ L ) → GL k ( C ) be a representation such that the ρ -twisted chain complex C ∗ ( S \ L ; ρ ) is acyclic, in which case we say the GL k ( C )-link L is acyclic. Then the ρ -twisted torsion τ ( L, ρ ) is the torsion of the ρ -twistedcohomology C ∗ ( S \ L ; ρ ).Usually when ρ has abelian image this is called the Reidemeister torsion . Whenthe image of ρ is nonabelian it is called the twisted torsion. We prefer to instead referto these cases as abelian and nonabelian torsions. As mentioned earlier, usually thetorsion is defined in terms of homology, but cohomology is better for our purposes,and they give the same torsion in the case of interest because there is a Poincar´eduality for SL ( C )-coefficients. Proposition 3.6.
Let ( L, ρ ) be a GL k ( C ) -link such that det(1 − ρ ( x i )) (cid:54) = 0 forevery meridian x i . Then (1) the complex H ∗ ( S \ L, ρ ) is acyclic, so the torsion τ ( L, ρ ) is a complexnumber defined up to ± det ρ , OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 19 (2) if ( L, ρ ) is the closure of a GL k ( C ) -braid β on n strands with det(1 − ρ ( y n )) nonsingular, we can compute the torsion as τ ( L, ρ ) = det(1 − B ( β ))det(1 − ρ ( y n )) where y n = x · · · x n is a path around all the punctures of D n , and (3) if ( L, ρ ) is an SL ( C ) -link there always exists such a braid β .Proof. (1) and (2) are standard results in the theory of torsions. The idea is thatwe use the basis coming from y ∗ , . . . , y ∗ n − for H and from y ∗ n for H , and thesebases give nondegenerate matrix τ -chains [22] for the complex, so they compute thetorsion. More details can be found in [6, Theorem 3.15]; that paper discusses twistedAlexander polynomials, which correspond with the torsion when the variables t i areall 1.The only novel (to our knowlege) claim is (3). Represent ( L, ρ ) as the closureof a SL ( C )-braid β on n strands which is an endomorphism of the color tuple( g , . . . , g n ), and write h n = g · · · g n for the total holonomy. Consider the coloredbraids β : ( g , · · · , g n ) → ( g , · · · , g n ) βσ n : ( g , · · · , g n , g n ) → ( g , · · · , g n , g n ) βσ n σ n +1 : ( g , · · · , g n , g n , g n ) → ( g , · · · , g n , g n , g n )Their closures are all ( L, ρ ), and they have total holonomies h n , h n g n , h n g n respectively. Recall that an element g ∈ SL ( C ) has 1 as an eigenvalue if and onlyif tr g = 2. Similarly, for elements of SL ( C ) we have the trace identitytr( h n g n ) + tr( h n ) = tr( h n g n ) tr( g n )But tr g n (cid:54) = 2, so at least one of tr( h n g n ) , tr( h n g n ) , or tr h n has trace not equalto 2. We conclude that at least one braid with closure ( L, ρ ) has nontrivial totalholonomy. (cid:3)
Taking the closure of a braid relates the complex C ∗ ( D n ; ρ ) to C ∗ ( S \ L ; ρ ) byadding a term in dimension 2, so it is reasonable to expect a relationship betweenthe torsion and the Burau representation. Notice that when the image of ρ lies inSL n the torsion is defined up to an overall sign.In general, any sufficiently nontrivial representation ρ will be acyclic. For exam-ple, this is the case if ρ ( x ) never has 1 as an eigenvalue for a meridian x . It is alsotrue for representations coming from the holonomy of a finite-volume hyperbolicstructure [18] 4. The algebra U i ( sl )4.1. Quantum sl at a fourth root of unity. Quantum sl is the algebra U q = U q ( sl ) over C [ q, q − ] with generators E, F, K ± and relations KE = q EKKF = q − F K [ E, F ] = ( q − q − )( K − K − ) We sometimes use the generator ˜ F = qKF instead of F . Notice that our conven-tions are slightly nonstandard (in particular, they differ from [5].) We want to view U q as a deformation of the algebra of functions on SL ( C ) ∗ , not a deformation ofthe universal enveloping algebra of sl . For this reason, we choose [ E, F ] as aboveinstead of the more common [
E, F ] = ( K − K − ) / ( q − q − ). U q is a Hopf algebra, with coproduct∆ E = 1 ⊗ E + E ⊗ K, ∆ F = K − ⊗ F + F ⊗ , ∆ K = K ⊗ K and antipode S ( E ) = − EK − , S ( F ) = − KF, S ( K ) = K − . The center of U is generated by the Casimir element˜Ω = EF + q − K + qK − We will mostly work with the normalization Ω = q ˜Ω.We consider the case where q is specialized to a primitive fourth root of unity i ,which is (cid:96) = 4 , r = 2 in [5]. The relations for U = U i ( sl ) are then KE = − EKKF = − F K [ E, F ] = 2 i ( K − K − )Specializing to a root of unity causes U to have a large central subalgebra Z := C [ K , K − , E , F ]The center Z of U is generated by Z and the Casimir Ω, subject to the relationΩ = ( K − K − ) − E F We can identify the closed points of Spec Z with the set of characters , that is alge-bra homomorphisms χ : Z → C . The characters form a group with multiplication χ χ ( x ) := χ ⊗ χ (∆( x )). In fact, this group is SL ( C ) ∗ : Proposition 4.1.
Let χ be a Z -character and set χ ( E ) = ε, χ ( F ) = ϕ/κ, χ ( K ) = κ The map sending χ to the group element (cid:18)(cid:18) κ ϕ (cid:19) , (cid:18) ε κ (cid:19)(cid:19) ∈ SL ( C ) ∗ is an isomorphism of algebraic groups Spec Z → SL ( C ) ∗ . The inverse χ − of acharacter is the character χS obtained by precomposition with the antipode. From now on we identify Z -characters and the corresponding points of SL ( C ) ∗ .The image of a character χ is the factorization of the matrix ψ ( χ ) := (cid:18) κ − εϕ κ − (1 − εϕ ) (cid:19) ∈ SL ( C )so this identification is compatible with the factorization of SL ( C ) in terms ofSL ( C ) ∗ . Here we have intentionally used the same symbol ψ for the defactorizationmaps SL ( C ) ∗ → SL ( C ) and Spec Z → SL ( C ). OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 21
Irreducible representations of U are parametrized by the spectrum of its center Z ,which is a quadratic extension of Z by the Casimir Ω. Suppose ˆ χ is a Z -character.Since ˆ χ (Ω ) = κ + (1 − εϕ ) κ − − ψ ( ˆ χ ) − Z -character is given by a Z -character (that is, a point of SL ( C ) ∗ )and a square root of the trace minus 2. In particular, we can identify Z -charactersas SL ( C ) ∗ elements a along with a square root ω of tr ψ ( a ) −
2, which we call a
Casimir value . (See also Definition 4.3.) We will see in § Weight modules for U .Definition 4.2. A U -weight module is a representation V of U on which the cen-tral subalgebra Z acts diagonalizably. Let χ be a Z -character, i.e. an algebrahomomorphism Z → C . We say a representation V of U has character χ if Z · v = χ ( Z ) · v for every Z ∈ Z and v ∈ V .Every irreducible weight module has a character by definition, and in generalany finite-dimensional weight module V decomposes as a direct sum V = (cid:77) χ V χ where V χ is the submodule on which Z acts by χ .Equivalently, given a character χ , we can consider the quotient algebra U /I χ where I χ is the ideal generated by the kernel of χ . A U -module has character χ ifand only if the structure map factors through U /I χ .Write C χ for the category of finite-dimensional U /I χ -representations, i.e. thecategory of U -weight modules with character χ . We will discuss how the categories C χ fit together into a graded category in the next section. Theorem 1.
We say that a Z -character χ is nonsingular if tr ψ ( χ ) (cid:54) = 2 . In thiscase: (1) C χ is semisimple, (2) the simple objects of C χ are all -dimensional and projective, and (3) isomorphism classes of simple objects are parametrized by the Casimir Ω ,which acts by a square root of tr ψ ( χ ) − .Proof. This is a special case of [5, Theorem 6.2]. The idea is to use a certainHamiltonian flow (the quantum coadjoint action of Kac-de Concini-Procesi) onSpec Z to reduce to the case χ ( E ) = χ ( F ) = 0. (cid:3) In particular, the character ε ( K ) = 1, ε ( E ) = ε ( F ) = 0 corresponding to theidentity matrix is singular. The category C ε is the category of modules of the smallquantum group, which is not semisimple. Irreducible modules.
We discuss the modules of Theorem 1 in more detail.
Definition 4.3.
Let χ be a nonsingular Z -character corresponding to the SL ( C ) ∗ element a , and let ω be a complex number with ω = χ (Ω ) = tr ψ ( χ ) −
2. Wewrite V ( χ, ω ) = V ( a, ω ) = V ( ˆ χ )for the irreducible module of dimension 2 with character χ on which the CasimirΩ acts by the scalar ω . We sometimes call these V ( ˆ χ ) nonsingular modules. Hereˆ χ is the extension of χ to Z given by ˆ χ (Ω) = ω . Proposition 4.4.
Let χ i be nonsingular characters with Casimir values ω i . If theproduct character χ · · · χ n is nonsingular, then n (cid:79) i =1 V ( χ i , ω i ) ∼ = V ( χ · · · χ n , ω ) ⊕ n − ⊕ V ( χ · · · χ n , − ω ) ⊕ n − where ω is a Casimir value for the product character.Proof. This is easy to check for n = 2, and the general case follows by induction. (cid:3) We think of the left-hand side as representing a colored braid on n strandswith colors ( χ , . . . , χ n ), so a path wrapping around the entire braid has holonomy χ · · · χ n . The proposition says that the corresponding tensor product of irrepsdecomposes in to an equal number of summands of each module with Z -character χ · · · χ n .The modules V ( χ, ω ) can be difficult to work with. One reason is that writ-ing down a basis requires diagonalizing the action of K , which has eigenvalues ± (cid:112) χ ( K ). Avoiding these problems is one reason to consider the graded double D of the category C of weight modules.Let χ be a nonsingular Z -character, with χ ( K ) = κ, χ ( E ) = ε, χ ( F ) = ϕ/κ and choose a Casimir value ω , that is a complex number satisfying ω + 2 = κ + (1 − εϕ ) /κ = tr ψ ( χ ) = tr (cid:18) κ − εϕ (1 − εϕ ) /κ (cid:19) The requirement that χ be nonsingular is equivalent to requiring ω (cid:54) = 0.Let V ( χ, ω ) be the corresponding irreducible 2-dimensional U -module. It is nothard to see that we can always choose an eigenvector | (cid:105) of K such that | (cid:105) , | (cid:105) := E | (cid:105) is a basis of V ( χ, ω ).First consider the case where εϕ (cid:54) = 0; this is generically true, since non-triangularmatrices are dense in SL ( C ). Then with respect to the basis | (cid:105) , | (cid:105) , the generatorsact by π ( K ) = (cid:18) µ − µ (cid:19) , π ( E ) = (cid:18) ε (cid:19) ,π ( F ) = (cid:18) − i ( ω + µ − µ − ) − i ( ω − µ + µ − ) /ε (cid:19) where µ is an arbitrarily chosen square root of κ . We can think of | (cid:105) and | (cid:105) asa weight basis. Since E and F act invertibly V ( χ, ω ) is sometimes called a cyclic module. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 23
The case εϕ = 0 is simpler. Then one (or both) of E, F act nilpotently, so V ( χ, ω ) is said to be semi-cyclic (or nilpotent. ) Suppose in particular that ϕ = 0(the case ε = 0 is similar) and choose an eigenvector | (cid:105) of K with F | (cid:105) = 0. Thenthe action of the generators is given by π ( K ) = (cid:18) µ − µ (cid:19) , π ( E ) = (cid:18) ε (cid:19) , π ( F ) = (cid:18) − iω (cid:19) where ω = µ − µ − . Notice that the choice of square root µ of κ is no longerarbitrary: if v ∈ Ker F , then we must have Kv = µ and ω = µ − µ − .4.4. Unit-graded representations.
In addition to the nonsingular modules, wewill need to consider certain modules corresponding to singular characters.
Example 4.5.
As usual for the category of representations of a Hopf algebra, thetensor unit is the vector space C , with the action of U given by the counit: ε ( K ) = 1 , ε ( E ) = ε ( F ) = 0 . This module is irreducible, with Z -character ε = ε | Z . The corresponding elementof SL ( C ) ∗ is the identity element, which we expect, since for a module V withcharacter χ , the character of ⊗ V ∼ = V should be ε · χ = χ .Because C is not semisimple, is not a projective U -module. It is not hard todescribe its projective cover, however. Definition 4.6. P is the 4-dimensional U -module described by K (cid:55)→ − − , E (cid:55)→ , F (cid:55)→ − i − i P has Z -character the counit ε , thought of as an algebra homomorphism Z → C .Note that P is indecomposable, but not irreducible, and that Ω does not actdiagonalizably on P .The action of the generators of U on P is best described diagrammatrically.There is a basis x, y , y , z of P with K · x = x, K · z = z, K · y i = − y i and with the action of E and F given by the diagram xy y z E FF E where missing arrows mean action by 0, e.g. E · y = 0. Proposition 4.7. (1) P is the projective cover (dually, the injective hull) ofthe tensor unit . (2) For any nonsingular Z -character χ with Casimir value ω , V ( χ, ω ) ⊗ V ( χ, ω ) ∗ ∼ = P
04 CALVIN MCPHAIL-SNYDER (3)
The vector spaces
Hom U ( P , ) and Hom U ( , P ) are one-dimensional. Theisomorphism in (2) takes ev V to a basis of Hom U ( P , ) and coev V to abasis of Hom U ( , P ) . Finally, we consider two more modules with character ε . Definition 4.8.
The parity module
Π is the 1-dimensional U -representation withaction K (cid:55)→ − , E (cid:55)→ , F (cid:55)→ P − := Π ⊗ P It is easy to describe P − : the action of U is the same, except that the sign of K is switched. The name “parity module” is because of the following proposition: Proposition 4.9.
For any admissible Z -character χ with Casimir ω , Π ⊗ V ( χ, ω ) ∼ = V ( χ, ω ) ⊗ Π ∼ = V ( χ, − ω ) . Similarly, we have V ( χ, ω ) ⊗ V ( χ, − ω ) ∗ ∼ = P − . Holonomy braiding.
We describe the braiding for U -weight modules. Un-like U q , the algebra U = U i is not quasitriangular. Instead, there is an outerautomorphism R : U ⊗ U → U ⊗ U [(1 + K − E ⊗ F ) − ]that satisfies Yang-Baxter equations(∆ ⊗ R ( u ⊗ v ) = R R (∆( u ) ⊗ v )(9) (1 ⊗ ∆) R ( u ⊗ v ) = R R ( u ⊗ ∆( v ))(10)and ( ε ⊗ R ( u ⊗ v ) = ε ( u ) v (1 ⊗ ε ) R ( u ⊗ v ) = ε ( v ) u where ∆ is the coproduct, ε the counit, and R ij means the action on the i th and j th tensor factors.In a quasitriangular Hopf algebra, R comes from conjugation by an elementcalled the R -matrix . This is not the case for U = U i , but there is a version of U q defined over formal power series in h (with q = e h ) which has an R -matrix. Theconjugation action of this element is still well-defined in the specialization q = i ,giving the outer automorphism R . For more details, see the paper [15].The braid action is given by applying R and then the flip τ ( x ⊗ y ) = y ⊗ x , so wewill mostly work with the automorphism ˇ R := τ R . ˇ R is the unique automorphismof U ⊗ U [ W − ] satisfying ˇ R ( E ⊗
1) = K ⊗ E ˇ R (1 ⊗ F ) = F ⊗ K − ˇ R ( K ⊗
1) = 1 ⊗ K − iKF ⊗ E Here ev and coev are part of the pivotal structure on the category of U -weight modules. SeeProposition 5.10 for details. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 25 and ˇ R (∆( u )) = ∆( u )for every u ∈ U , where W := 1 + K F ⊗ K − E ∈ Z ⊗ Z . The localization at W corresponds to the fact that the SL ( C ) ∗ biquandle is par-tially defined.We call ˇ R a holonomy braiding because it is compatible with the biquandleSL ( C ) ∗ in the following sense. The action of ˇ R on Z ⊗ Z [ W − ] is given byˇ R ( K ⊗
1) = (1 ⊗ K ) W ˇ R (1 ⊗ K ) = ( K ⊗ W − ˇ R ( E ⊗
1) = K ⊗ E ˇ R (1 ⊗ E ) = E ⊗ K + E ⊗ (1 − K W − )ˇ R ( F ⊗
1) = K − ⊗ F + (1 − K − W − ) ⊗ F ˇ R (1 ⊗ F ) = F ⊗ K − Consider a SL ( C ) ∗ -colored braid generator σ : ( χ , χ ) → ( χ (cid:48) , χ (cid:48) ), thinking ofSL ( C ) ∗ elements as Z -characters. Then Proposition 4.10. ˇ R is compatible with the SL ( C ) ∗ biquandle in the sense that ( χ (cid:48) ⊗ χ (cid:48) ) ˇ R = χ ⊗ χ In particular, ˇ R descends to a homomorphism of algebras ˇ R : U /I χ ⊗ U /I χ → U /I χ (cid:48) ⊗ U /I χ (cid:48) , and this correspondence extends to the enhanced SL ( C ) ∗ biquandle.Proof. Write χ i ( K ) = κ i , χ i ( E ) = ε i , χ i ( F ) = ϕ i /κ i , and similarly for theimages χ (cid:48) i under the braiding. Then, for example,( χ ⊗ χ )( K ⊗
1) = κ is, via (4), equal to ( χ (cid:48) ⊗ χ (cid:48) )((1 ⊗ K ) W ) = κ (cid:48) + ϕ (cid:48) ε (cid:48) One can check similar relations for the other generators of Z ⊗ Z . The formalinversion of W is not an issue, because( χ ⊗ χ )( W ) = 1 + ϕ ε κ − is nonzero exactly when the SL ( C ) ∗ colors ( χ , χ ) are admissible.Finally, since ˇ R (Ω ⊗
1) = 1 ⊗ Ω , ˇ R (1 ⊗ Ω) = Ω ⊗ (cid:3) The automorphism ˇ R acts on the algebras U /χ , but we want a braiding on themodules V ( χ ). Such a braiding is a family of maps intertwining ˇ R in the followingsense: Definition 4.11.
Let χ , χ be nonsingular Z -characters such that ( χ , χ ) = B ( χ , χ ) exists (equivalently, such that the SL ( C ) ∗ -colored braid σ : ( χ , χ ) → ( χ , χ ) is admissible.) For each i , let X χ i be a module with character χ i . We saya map c : X χ ⊗ X χ → X χ ⊗ X χ of U -modules is a holonomy braiding if for every u ∈ U ⊗ U and x ∈ X χ ⊗ X χ , wehave c ( u · x ) = ˇ R ( u ) · c ( x )Since ˇ R preserves the coproduct, a holonomy braiding is automatically a mapof U -modules. Proposition 4.12.
Let χ i , i = 1 , . . . , be characters as in Definition 4.11, andchoose Casimir values ω , ω for χ , χ . Then there is a nonzero holonomy braiding c : V ( χ , ω ) ⊗ V ( χ , ω ) → V ( χ , ω ) ⊗ V ( χ , ω ) unique up to an overall scalar. Here V ( χ i , ω i ) is the irreducible U -module with character χ i and Casimir ω i . Proof.
We follow the discussion proceeding [5, Theorem 6.2]. Write ˆ χ i for the Z -character extending χ i by ˆ χ i (Ω) = ω i , setting ω = ω , ω = ω . For each i , thealgebra U /I ˆ χ i (where I ˆ χ i is the ideal generated by the kernel of ˆ χ i ) is isomorphic to the C -endomorphism algebra Mat ( C ) of V ( ˆ χ i ) = V ( χ i , ω i ). We see that the automor-phism ˇ R induces an automorphism of matrix algebrasMat ( C ) ⊗ Mat ( C ) → Mat ( C ) ⊗ Mat ( C )equivalently, an automorphism of matrix algebrasˇ R : Mat ( C ) → Mat ( C )By linear algebra, any such automorphism is inner, given by ˇ R ( X ) = cXc − forsome invertible matrix c , which is unique up to an overall scalar. The matrix c gives the holonomy braiding with respect to the bases of V ( χ i , ω i ) implicit in theisomorphisms U /I ˆ χ i ∼ = Mat ( C ). (cid:3) We will see later that the braidings c = c χ ,χ fit together into a representation ofthe category ˆ B (SL ( C )) ∗ . However, the ambiguous normalization of the braidingsmeans that this is only a projective representation.5. Representation categories for U i ( sl )In this section we discuss three related categories constructed from represen-tations of the algebra U = U i ( sl ), the first of which is C , the category of well-behaved U -modules. We also introduce an opposite version C of C and a category D = C (cid:2) SL ( C ) ∗ C we can think of as being a graded version of the quantum doubleof C .These categories are all graded by the group SL ( C ) ∗ and have a holonomybraiding generalizing the usual notion of braiding. However, for C and C we areonly able to define this braiding projectively. One motivation for introducing thecategory D is to eliminate this ambiguity in the braiding. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 27
The category C of weight modules.Definition 5.1. C is the category of finite-dimensional U -weight modules and U -module intertwiners. It is an SL ( C ) ∗ -graded category C = (cid:77) χ ∈ SL ( C ) ∗ C χ where C χ is the subcategory of modules with character χ , equivalently the subcate-gory of (finite-dimensional) representations of U /I χ . For homogeneous objects, wewrite | V | for the degree (also called the grading or coloring) of V .The direct sum above means that objects of C decompose as direct sums V = (cid:76) χ ∈ SL ( C ) ∗ V χ , where V χ is the submodule on which Z acts by χ . Morphisms of C are graded in the sense that for homogeneous objects V and W , Hom C ( V, W ) isnonzero only when | V | = | W | .As the category of representations of a Hopf algebra, C is a rigid monoidalcategory, and the tensor product and duality are compatible with the grading inthe sense that if | V | = χ , | V | = χ , then | V ⊗ V | = χ χ , | V ∗ | = χ − . The grading rules for tensor products and duals follow from Proposition 4.1.5.2.
The opposite category of C . C is the opposite category to C ; that is, thecategory with the inverse grading, opposite tensor product, and inverse braiding to C . We explain this in more detail. Definition 5.2. C is an SL ( C ) ∗ -graded tensor category with the same objects andmorphisms as C . The grading is inverted in the sense that if V ∈ C and | V | = χ , V is a U -weight module with character χ − . The tensor product ⊗ is opposite to thetensor product ⊗ of C : V ⊗ W = V ⊗ C W as a vector space, with the U -modulestructure given by X · ( v ⊗ w ) = ∆ op ( X ) · ( v ⊗ w )Taking the opposite braiding requires more care. Let ˇ R ∗ be the opposite, inversemorphism to ˇ R . That is, Definition 5.3.
Set W = 1 + F ⊗ E . ˇ R ∗ is the unique automorphism of U ⊗U [ W − ] satisfying ˇ R ∗ ( E ⊗ K ) = 1 ⊗ E ˇ R ∗ ( K − ⊗ F ) = F ⊗ R ∗ ( K ⊗ − E ⊗ iKF ) = 1 ⊗ K and ˇ R ∗ (∆ op ( u )) = ∆ op ( u )for every u ∈ U .These equations are obtained by swapping tensor factors in the rules for ˇ R − .By making these changes, we can obtain holonomy braidings for the simple objectsof C . Remark 5.4.
The last rule is equivalent toˇ R ∗ ( K − ⊗
1) = 1 ⊗ K − − iF ⊗ K − E Proposition 5.5.
Let χ , χ be nonsingular characters such that B ( χ , χ ) =( χ , χ ) exists. (1) B − ( χ − , χ − ) exists and is equal to ( χ − , χ − ) . (2) ˇ R ∗ defines an algebra automorphism U /I χ − ⊗ U /I χ − → U /I χ − ⊗ U /I χ − (3) Choose Casimir values ω , ω for χ , χ . These are still Casimir values for χ − and χ − , respectively, and there is a nonzero holonomy braiding c : V ( χ − , ω ) ⊗ V ( χ − , ω ) → V ( χ − , ω ) ⊗ V ( χ − , ω ) unique up to an overall scalar, satisfying c ( X · v ) = ˇ R ∗ ( X ) · c ( v ) for every v ∈ V ( χ − , ω ) ⊗ V ( χ − , ω ) and X ∈ U ⊗ U .Proof. To prove (1), recall the derivation of the biquandle relations: from the col-ored braid diagram a a a a we can write down the equations a − a − = a − a − a − a +2 = a +4 a − a +1 a +2 = a +4 a +3 For admissible a , a these equations have a unique solution ( a , a ) = B ( a , a ),which defines the biquandle structure on SL ( C ) ∗ . On the other hand, from themirror diagram b b b b we have b − b − = b − b − b +2 b − = b − b +4 b +2 b +1 = b +3 b +4 Notice that we have reversed the numbering of the ends in the second diagram; thiscorresponds to taking the opposite coproduct on U in the tensor product ⊗ for C ,hence the opposite product in the group Spec Z . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 29
The equations in the b variables are exactly the same as those in the a variablesif b i = a − i as elements of SL ( C ) ∗ for all i . Furthermore, because ( S ⊗ S )( W ) = W ,the image of W in U /I χ − ⊗U /I χ − is the same as the image of W in U /I χ ⊗U /I χ ,so the admissibility conditions are still compatible with the localization in thedefinition of ˇ R ∗ . This proves (1). (2) and (3) now follow from essentially the sameproofs as Propositions 4.10 and 4.12. (cid:3) Remark 5.6.
The holonomy braiding c constructed above is equivalent to a map c : V ( χ , ω ) ∗ ⊗ V ( χ , ω ) ∗ → V ( χ , ω ) ∗ ⊗ V ( χ , ω ) ∗ via the U -module isomorphisms V ( χ, ω ) ∗ ∼ = V ( χ − , ω ) . The graded double of C . We want to define a category D which is a certainsubcategory of the Deligne tensor product C (cid:2) C of C and ¯ C . For a general discussionof the Deligne tensor product, see [8, § H (cid:2) Rep H is equivalent to Rep( H ⊗ H ) for Hopf algebras H , H ,under some finiteness hypotheses that apply here. Definition 5.7.
A representation V of U ⊗ C U is a weight module if Z ⊗ C Z ⊆U ⊗ C U acts diagonalizably on V . D is the category of finite-dimensional weightmodules of U ⊗ C U which are locally homogeneous in the sense that for every v ∈ V and Z ∈ Z , ( Z (cid:2) · v = (1 (cid:2) S ( Z )) · v D is a SL ( C ) ∗ -graded category: an object of degree χ is a weight module V suchthat, for every v ∈ V and Z ∈ Z ,( Z (cid:2) · v = (1 (cid:2) S ( Z )) · v = χ ( Z ) v Here we have inserted the antipode to reflect the fact that an object of degree χ in C is a representation with character χ − . In the above expressions, the symbol (cid:2) means ⊗ C , but we prefer (cid:2) to emphasize the external tensor product of categoriesversus the internal tensor product on C , which we will continue to denote by ⊗ .Similarly, going forward we write U (cid:2) U to mean U ⊗ C U . Example 5.8.
Let V ( χ, ω ) be a nonsingular irreducible in C (that is, one of the2-dimensional modules of Theorem 1.) Then the U (cid:2) U -module W ( χ, ω ) := V ( χ, ω ) (cid:2) V ( χ, ω ) ∗ is a simple object of degree χ in D . (Again (cid:2) means ⊗ C , but we use this notationto emphasize the connection with the Deligne tensor product.)The tensor product in D is obtained in the obvious way from the tensor prod-ucts of C and C . In particular, for the simple objects of the previous example,abbreviating V i = V ( χ i , ω i ) we have( V (cid:2) V ∗ ) ⊗ ( V (cid:2) V ∗ ) = ( V ⊗ V ) (cid:2) ( V ∗ ⊗ V ∗ )The notion of holonomy braiding extends in the obvious way to D : a holonomybraiding is a map c intertwining the automorphism ˇ R (cid:2) ˇ R ∗ of ( U (cid:2) U ) ⊗ ( U (cid:2) U )which acts on the first and third tensor factors by ˇ R and second and fourth by ˇ R ∗ . Proposition 5.9.
The holonomy braidings c − , − , c − , − for C and C give a projectiveholonomy braiding c ˆ χ , ˆ χ (cid:2) c ˆ χ , ˆ χ : W ( ˆ χ ) ⊗ W ( ˆ χ ) → W ( ˆ χ ) ⊗ W ( ˆ χ ) of D , where W is the family of irreducible modules of Example 5.8. Modified traces.
As discussed in the next section, we can use the holonomybraidings to obtain representations of B (SL ( C )) ∗ in the categories C , C , and D .To get link invariants we want to take the closures of these braids, which requires apivotal structure. However, there is a complication: the link invariants obtained inthis way are uniformly zero. To fix this problem, we can use the theory of modifiedtraces. Proposition 5.10. C is a pivotal category with pivot K − ∈ U . That is, for anobject V of C with basis { v j } and dual basis { v j } the coevaluation (creation, birth) and evaluation (annihilation, death) morphisms are given by coev V : C → V ⊗ V ∗ , (cid:55)→ (cid:80) j v j ⊗ v j coev V : C → V ∗ ⊗ V, (cid:55)→ (cid:80) j v j ⊗ Kv j ev V : V ⊗ V ∗ → C , v ⊗ f (cid:55)→ f ( K − v )ev V : V ∗ ⊗ V → C , f ⊗ v (cid:55)→ f ( v ) so that for any morphism f : V → V , the quantum trace is the complex number tr f := ev V ( f ⊗ id V ∗ ) coev V where we identify linear maps C → C with elements of C . The quantum dimension of V is tr(id V ) . Furthermore, C is spherical: the right trace above agrees with theleft trace ev V (id V ∗ ⊗ f )coev V Proof.
This works because the square of the antipode of U is given by conjugationwith the grouplike element K − . It is not hard to check directly that the left andright traces agree. (cid:3) It is clear that this pivotal structure extends to the opposite category C , andthat K − (cid:2) K − is a pivot for U (cid:2) U , so that D is also pivotal. Furthermore, Corollary 5.11.
The quantum traces on C , C , and D are compatible in the sensethat if V (cid:2) V is an object of D and f ∈ End C ( V ) , g ∈ End C ( V ) , then tr( f (cid:2) g ) = tr( f ) tr( g )Unfortunately, because C is not semisimple, most of the quantum dimensionsvanish, including those of all the nonsingular modules of Theorem 1. We can seethis directly. Suppose V ( χ, ω ) is such a module. Then χ ( K ) = κ for some κ ∈ C ,so K has eigenvalues ± κ / , and in a K -eigenbasis we can computedim( V ( χ, ω )) = tr (cid:18) κ / − κ − / (cid:19) = 0Then the quantum dimension of every irrep is zero, so the value of any nontrivialclosed diagram (in particular, any link) is zero.This problem can be solved by the introduction of a modified trace. We explainthe basic idea before giving a formal definition. Let f : V ⊗ · · · ⊗ V n → V ⊗ · · · ⊗ V n OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 31 be an endomorphism of C . We can take the partial quantum trace on the right-handtensor factors V ⊗ · · · ⊗ V n to obtain a mapptr rq ( f ) : V → V If V is irreducible, ptr rq ( f ) = x id V for some scalar x , so we say that f has modifiedtrace t ( f ) := x d ( V )where d ( V ) is the renormalized dimension of V .Of course, the trace will depend on the choice of renormalized dimensions. Inthe case V = · · · = V n , such as when defining the abelian Conway potential, thechoice of renormalized dimension only affects the normalization of the invariant.However, when the modules can differ, the numbers d ( V ) must be chosen carefullyto insure that we obtain a link invariant. Theorem 2.
Let
Proj be the subcategory of C of projective U -weight modules. Proj admits a nontrivial modified trace t , unique up to an overall scalar. That is, forevery projective object V of C there is a linear map t V : End C ( V ) → C , these maps are cyclic in the sense that for any f : V → W and g : W → V we have t V ( gf ) = t W ( f g ) , and they agree with the partial quantum traces in the sense that if V ∈ Proj and W is any object of C , then for any f ∈ End C ( V ⊗ W ) , we have t V ⊗ W ( f ) = t V (ptr rW ( f )) where tr rW is the partial quantum trace on W .This trace corresponds to the renormalized dimensions d ( V ( ˆ χ )) := t (id V (ˆ χ ) ) = 1ˆ χ (Ω) where V ( ˆ χ ) is the irreducible U -module with Z -character ˆ χ . We usually omit the subscript on t , and we have chosen a different normalizationof the dimensions than in [5] which is more natural for q = i . Notice that therenormalized dimensions are gauge-invariant, in the sense that if ψ ( ˆ χ ) is conjugateto ψ ( ˆ χ ), then d ( V ( ˆ χ )) = d ( V ( ˆ χ ))Since ˆ χ (Ω) = tr ψ ( ˆ χ ) −
2, it is not surprising that χ (Ω) is conjugation-invariant,i.e. gauge-invariant. Similarly, d ( V ( ˆ χ )) = d ( V ( ˆ χ ) ∗ ) for all nonsingular ˆ χ . Proof.
See [5, § (cid:3) Corollary 5.12.
Let
Proj( C ) be the subcategory of C of projective U -modules. Proj( C ) admits a nontrivial modified trace t , unique up to an overall scalar. Thistrace corresponds to the renormalized dimensions d ( V ( ˆ χ )) := t (id V (ˆ χ ) ) = 1ˆ χ (Ω) Proof.
This is an immediate consequence of the definition of C as the opposite cate-gory to C . The fact that the renormalized dimensions are the same is a consequenceof the fact that S (Ω) = Ω. (cid:3) Theorem 3.
Let
Proj( D ) be the subcategory of projective U (cid:2) U -modules of D . Proj( D ) admits a nontrivial modified trace with renormalized dimensions d ( V ( χ, ω ) (cid:2) V ( χ, ω ) ∗ ) = 1 ω which is compatible with the traces on C and C . That is, let W be a projective objectof C and W a projective object of C . Then for any endomorphisms f : W → W , g : W ⊗ W , t ( f (cid:2) g ) = t ( f ) t ( g ) with f (cid:2) g the obvious endomorphism of W (cid:2) W .Proof. This theorem is easy to prove using the techniques of [10]. For completenesswe include the proof in Appendix A. (cid:3) Quantum holonomy invariants
In this section, we use the holonomy braidings of the previous section to de-fine three closely related representations of the enhanced SL ( C ) ∗ -colored braidgroupoid ˆ B (SL ( C )) ∗ . Via the modified traces, these give link invariants.6.1. Representations of the colored braid groupoid.Definition 6.1.
Let ˆ χ , ˆ χ be colors of the enhanced SL ( C ) ∗ biquandle, i.e. Z -characters. A representation of ˆ B (SL ( C )) ∗ valued in a monoidal, C -linear category C is a collection of objects { V ˆ χ } of C and a collection of invertible morphisms c ˆ χ , ˆ χ : V ˆ χ ⊗ V ˆ χ → V ˆ χ ⊗ V ˆ χ which exist whenever ( ˆ χ , ˆ χ ) = B ( ˆ χ , ˆ χ ) is defined, and which satisfy the coloredReidemeister moves (i.e. the relations of the groupoid ˆ B (SL ( C )) ∗ .) If instead the c − , − satisfy the colored Reidemeister moves up to multiplication by some subgroupΓ ⊆ C × , we say that ( { V − } , { c − , − } ) is a representation with indeterminacy Γ . That is, a representation is a choice of object V ˆ χ to assign to a strand colored byˆ χ , along with holonomy braidings among these objects. A representation valued in C gives a monoidal functor ˆ B (SL ( C ) ∗ ) → C , via the assignments( ˆ χ , · · · , ˆ χ n ) (cid:55)→ V ˆ χ ⊗ · · · ⊗ V ˆ χ n and ( σ : ( ˆ χ , ˆ χ ) → ( ˆ χ , ˆ χ )) (cid:55)→ c ˆ χ , ˆ χ We usually refer to the representation via this functor.
Proposition 6.2.
Setting F ( ˆ χ ) = V ( ˆ χ ) along with the holonomy braidings of Proposition 4.12 give a representation F : ˆ B (SL ( C )) ∗ → C , Furthermore, the holonomy braidings can be normalized so that F has indeter-minacy (cid:104) i (cid:105) = { , i, − , − i } . Similarly, by setting F ( ˆ χ ) = V ( ˆ χ ) ∗ and using the OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 33 holonomy braidings on C we obtain a representation taking values in C with inde-terminacy (cid:104) i (cid:105) . Our choice of normalization is somewhat ad-hoc, and is chosen to agree with thedoubled representation T . This issue will mostly likely be clarified by the explicitformulas for the braiding computed in [17]. Proof.
We prove the theorem for F ; the proof for F is essentially identical.Recall that by Proposition 4.12 the holonomy braidings exist and are unique upto a scalar. By Proposition 4.10 and the Yang-Baxter relations (9, 10), they satisfythe colored Reidemeister moves, so we only need to deal with the normalizations.In [5], the authors fix this normalization by requiring det( c − , − ) = 1. Scaling thedeterminant of a 4 × (cid:104) i (cid:105) indeterminacy.We choose a different normalization. Let σ : ( χ , χ ) → ( χ (cid:48) , χ (cid:48) )be a braid generator between admissible enhanced characters; for notational sim-plicity we denote thsese χ instead of ˆ χ . We normalize c = c ˜ χ , ˜ χ so thatdet c = f ( χ (cid:48) , χ (cid:48) ) = χ (cid:48) ( K − ) χ (cid:48) ( K ) + χ (cid:48) ( F ) χ (cid:48) ( E ) . Notice that f ( χ (cid:48) , χ (cid:48) ) is nonzero exactly when σ is admissible.We must show that our choice of scaling factor f is compatible with coloredReidemeister moves. The RI move is not relevant to braids. For the RII move,consider ( χ , χ ) σ −→ ( χ (cid:48) , χ (cid:48) ) σ − −−→ ( χ , χ ) . It is not hard to show that f ( χ (cid:48) , χ (cid:48) ) f ( χ , χ ) = 1The RIII move is similar but more complicated. Consider the product of coloredbraids ( χ , χ , χ ) σ −→ ( χ , χ , χ ) σ −→ ( χ , χ , χ ) σ −→ ( χ , χ , χ )which has determinant f ( χ , χ ) f ( χ , χ ) f ( χ , χ ) . We need to show that this agrees with( χ , χ , χ ) σ −→ ( ˜ χ , ˜ χ , ˜ χ ) σ −→ ( ˜ χ , ˜ χ , ˜ χ ) σ −→ ( ˜ χ , ˜ χ , ˜ χ )which has determinant f ( ˜ χ , ˜ χ ) f ( ˜ χ , ˜ χ ) f ( ˜ χ , ˜ χ ) . Notice that ( ˜ χ , ˜ χ , ˜ χ ) = ( χ , χ , χ )follows from the biquandle being well-defined. Checking that f ( χ , χ ) f ( χ , χ ) f ( χ , χ ) = f ( ˜ χ , ˜ χ ) f ( ˜ χ , ˜ χ ) f ( ˜ χ , ˜ χ ) . is a tedious but straightforward computation. (cid:3) This is essentially showing that f is a biquandle 2-cocycle in the sense of [14]. The indeterminacy of this representation is why the invariant ∇ is only definedup to a fourth root of unity. We can use a “doubled” version of F to eliminate thisambiguity. Proposition 6.3.
The assignment T ( ˆ χ ) = V ( ˆ χ ) (cid:2) V ( ˆ χ ) ∗ and the holonomy braidings for D give a representation T : ˆ B (SL ( C )) ∗ → D with no scalar ambiguity.Proof. The key step is the following lemma, which will also be useful later.
Lemma 6.4.
Let a = ( ˆ χ , . . . , ˆ χ n ) be a tuple of nonsingular Z -characters. Thereis a family v ( ˆ χ , . . . , ˆ χ n ) ∈ T ( ˆ χ , . . . , ˆ χ n ) of vectors which are braiding-invariant in the sense that for every admissible braid β : a → b , T ( β )( v ( a )) = αv ( b ) for some nonzero scalar α .Proof. The idea is to consider the U (cid:2) U -modules W ( ˆ χ i ) appearing in the image of T as U -modules. Then, writing ⊗ for the product of U -modules, V ( ˆ χ i ) ⊗ V ( ˆ χ i ) ∼ = P where P is the module of Definition 4.6. W ( ˆ χ i ) is not the same as P , but we canstill exploit this similarity to simplify our computations. P is an indecomposable but reducible module, and can be characterized as theinjective hull of the tensor unit; the inclusion is the coevaluation mapcoev V i : → V ( ˆ χ i ) ⊗ V ( ˆ χ i ) ∗ ∼ = P whose image is the vector z := | (cid:105) ⊗ (cid:104) | + | (cid:105) ⊗ (cid:104) | writing | (cid:105) , | (cid:105) for the usual basis of V ( ˆ χ i ) (see § (cid:104) | , (cid:104) | for the dual basisof V ( ˆ χ i ) ∗ .The choice of inclusion map above fixes a canonical U -module isomorphism P → V ( ˆ χ ) ⊗ V ( ˆ χ ) for every nonsingular ˆ χ . Forgetting the U -module structure, we havea family of vector space isomorphisms f ˆ χ : P → V ( ˆ χ ) (cid:2) V ( ˆ χ ) ∗ and we define the invariant vectors by v ( ˆ χ , . . . , ˆ χ n ) := ( f ˆ χ ⊗ · · · ⊗ f ˆ χ n )( z ⊗ · · · ⊗ z ) . We now need to prove that they are invariant.It is enough to prove invariance in the case of a braid generator σ : W ( ˆ χ ) ⊗ W ( ˆ χ ) → W ( ˆ χ ) ⊗ W ( ˆ χ )where the ˆ χ i are nonsingular Z -characters such that B ( ˆ χ , ˆ χ ) = ( ˆ χ , ˆ χ ). We needto show that ( c (cid:2) c )( v ( ˆ χ , ˆ χ )) = αv ( ˆ χ , ˆ χ ) OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 35 for some nonzero α , where c , c are the holonomy braidings for V ( ˆ χ ) ⊗ V ( ˆ χ ) and V ( ˆ χ ) ∗ ⊗ V ( ˆ χ ) ∗ , respectively, and by c (cid:2) c we mean the operator acting on thetensor product V ( ˆ χ ) (cid:2) V ( ˆ χ ) ∗ ⊗ V ( ˆ χ ) (cid:2) V ( ˆ χ ) ∗ by c in factors 1 , c in factors 2 , γ = K (cid:2) K ⊗ K (cid:2) K − γ = K (cid:2) K − ⊗ (cid:2) − K (cid:2) ⊗ (cid:2) γ = E (cid:2) K ⊗ (cid:2) K + 1 (cid:2) E ⊗ (cid:2) Kγ = 1 (cid:2) K − ⊗ (cid:2) F − (cid:2) K − ⊗ F (cid:2) K − It is not hard to check that the kernel of these operators acting on W ( ˆ χ ) ⊗ W ( ˆ χ )is spanned by v ( ˆ χ , ˆ χ ). Hence by the definition of holonomy braiding0 = ( c (cid:2) c )( γ k · v ( ˆ χ , ˆ χ )) = ( ˇ R (cid:2) ˇ R ∗ )( γ k ) · ( c (cid:2) c )( v ( ˆ χ , ˆ χ ))for k = 0 , , , R and ˇ R ∗ that the images γ (cid:48) k = ( ˇ R (cid:2) ˇ R ∗ )( γ k ) are γ (cid:48) = K (cid:2) K ⊗ K (cid:2) K − γ (cid:48) = (1 ⊗ K − iKF ⊗ E ) (cid:2) (1 ⊗ K − − iF ⊗ K − E ) − ( K + K F ⊗ E ) (cid:2) (1 ⊗ γ (cid:48) = K (cid:2) K ⊗ E (cid:2) K + 1 (cid:2) ⊗ (cid:2) Eγ (cid:48) = 1 (cid:2) F ⊗ (cid:2) − F (cid:2) K − ⊗ K − (cid:2) K − To clarify the expressions in γ (cid:48) , consider the example( X ⊗ Y ) (cid:2) ( Z ⊗ W ) = X (cid:2) Z ⊗ Y (cid:2) W. We can compute that the kernel of the operators γ (cid:48) k acting on W ( ˆ χ ) ⊗ W ( ˆ χ )is spanned by v ( ˆ χ , ˆ χ ). Therefore ( c (cid:2) c )( v ( ˆ χ , ˆ χ )) must be proportional to v ( ˆ χ , ˆ χ ) as claimed. (cid:3) Given the lemma, we can choose the holonomy braidings to preserve the v ( − ).This gives a braid groupoid representation with no scalar ambiguity. (cid:3) Link invariants from braid groupoid representations.
By taking themodified traces of the representations constructed in the previous section we ob-tain link invariants. For simplicity (and to emphasize the connection with the Buraurepresentation) we focus on colored braids, but it is possible to construct represen-tations of more general colored tangle categories. These tangle representations aredeveloped further in [5].
Definition 6.5. A Markov trace on ˆ B (SL ( C )) ∗ is a family of functionst = t a : Hom ˆ B (SL ( C )) ∗ ( a, a ) → C for every object a = ( ˆ χ , . . . , ˆ χ n ) of ˆ B (SL ( C )) ∗ , which satisfy(1) conjugation invariance : for every β : a → a such that σ i βσ − i is admissible,t( σ i βσ − i ) = t( β ) , (2) and the Markov property: let β : a → a be a colored braid on n strands,and set a (cid:48) = ( ˆ χ , . . . , ˆ χ n +1 ). Then whenever the braid βσ n is an admissibleendomorphism, t( βσ n ) = θ ˆ χ n +1 t( β )where θ ˆ χ n +1 is a complex number, the twist of ˆ χ n +1 .When the twists are not 1, we obtain framed link invariants, not link invariants.Our representations do not depend on the framing, like the torsion. Proposition 6.6.
Let ( L, ρ, ω ) be an enhanced SL ( C ) -link which can be repre-sented as the closure of a braid β in ˆ B (SL ( C )) ∗ , and let t be a Markov trace on ˆ B (SL ( C )) ∗ with all twists . Then the complex number t( β ) is an invariant of ( L, ρ, ω ) . In particular, it does not depend on the choice of representative braid β .Proof. The usual Markov theorem says that any two representatives β, β (cid:48) of a link L can be related by a sequence of conjugation moves β (cid:55)→ σ i βσ − i and stabilizationmoves β (cid:55)→ βσ n (where β is a braid only involving the first n strands). This theoremextends in the obvious way to colored braids (see [11, Proposition 4.2] for detailsin a related case) and the proposition is an immediate consequence. (cid:3) Definition 6.7.
Let (
L, ρ, ω ) be an enhanced SL ( C )-link which can be representedas the closure of a braid β in ˆ B (SL ( C )) ∗ . Define ∇ ( L, ρ, ω ) := t ( F ( β )) ∇ ( L, ρ, ω ) := t (cid:0) F ( β ) (cid:1) T ( L, ρ, ω ) := t ( T ( β ))where t is the modified trace on the appropriate category.Not every SL ( C )-link can be represented as such a closure, because the im-age of ρ could lie outside the domain of the partially-defined defactorization mapSL ( C ) → SL ( C ) ∗ . However, every SL ( C )-link is gauge-equivalent to one ad-mitting such a representation. We will show in § ∇ , ∇ , and T are gauge-invariant, so they become well-defined invariants of all enhanced SL ( C )-links, bysetting ∇ ( L, ρ, ω ) = ∇ ( L, ρ g , ω )where ρ g ( − ) = gρ ( − ) g − is a representation conjugate to ρ for which ( L, ρ ) admitsa presentation as an SL ( C ) ∗ -braid closure. Theorem 4. T is an invariant of enhanced SL ( C ) -links which admit this presen-tation. ∇ and ∇ are similarly invariants up to a power of i .Proof. Lemma 6.8. t F , t F , and t T have writhe .Proof of the lemma. We prove this for t F ; t F is similar, and then t T follows.Using the notation of Definition 6.5, suppose that β : a → a and βσ n : a (cid:48) → a (cid:48) areboth endomorphisms. By Lemma 2.11 and the computations in § χ n = ˆ χ n +1 = ˆ χ , whereˆ χ ( K ) = µ , ˆ χ ( E ) = ˆ χ ( F ) = 0 , ˆ χ (Ω) = µ − µ − for some µ ∈ C \ { , , − } . Write | (cid:105) , | (cid:105) for the basis of V ( ˆ χ ) as in § c : V ( χ ) ⊗ V ( χ ) → V ( χ ) ⊗ V ( χ ). OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 37
With respect to the basis | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) (where | ij (cid:105) = | i (cid:105) ⊗ | j (cid:105) ) the matrix of c is µ − − (cid:0) µ − µ − (cid:1) − µ normalized so that det c = f ( ˆ χ, ˆ χ ) = 1, where f is the normalization factor inProposition 6.2.Because the renormalized trace is compatible with the partial trace, we cancompute t ( F ( βσ n )) = t (ptr n +1 ( F ( βσ n )))where by ptr n +1 we mean the trace on the ( n + 1)th (bottommost) strand. Butbecause β only involves the first n strands, we haveptr n +1 ( F ( βσ n )) = F ( β ) ptr n +1 ( F ( σ n ))and since F ( σ n ) = id V (ˆ χ ) ⊗ · · · ⊗ id V (ˆ χ n − ) ⊗ c it suffices to compute the partial trace of c on the second factor. The matrix of thelinear map (1 ⊗ K − ) c is µ − (cid:0) − µ − (cid:1) µ − µ − and taking the partial trace on the second tensor factor gives the endomorphism (cid:18) (cid:19) of V ( ˆ χ n ). Hence t ( F ( β ) ptr n +1 ( F ( σ n ))) = t ( F ( β ) id V (ˆ χ ) ⊗ · · · ⊗ id V (ˆ χ n ) ) = t ( F ( β ))as claimed. (cid:3) By Theorem 2, its corollary, and Proposition 6.2, t F and t F are Markov traces(up to sign). Similarly, by Theorem 3 and Proposition 6.3, t T is a Markov trace. Bythe lemma, these traces have writhe 1, so they give framing-independent invariants. (cid:3) ∇ is essentially the r = 2 case of the invariant F (cid:48) of [5]. However, we have chosena different normalization to match the doubled invariant T and the torsion τ . Theorem 5.
For any ( L, ρ, ω ) as above, ∇ ( L, ρ, ω ) ∇ ( L, ρ, ω ) = T ( L, ρ, ω ) up to an overall power of i .Proof. The braiding for T is the tensor product of the braidings for F and F ,so if we can show that the normalizations of the braidings are compatible, wecan use the compatibility of the renormalized traces from Theorem 3 to show theinvariants agree. It suffices to show the normalizations match for braid generators.Actually, we need only to show they match up a fourth root of unity because of theindeterminacy of F and F . Let σ i : a → b be a colored braid generator acting on strands i, i + 1 and write b = ( . . . , χ i , χ i +1 , . . . ). We show in § T ( a ) , T ( b ) such that the matrix of T ( σ i ) is (cid:16)(cid:94) B ⊗ id C (cid:17) for a matrix B coming from the action on the multiplicity space. id C appearsbecause this multiplicity space keeps track of 4-dimensional modules. Explicitly, B is the matrix I i − ⊕ /κ i − ϕ i /κ i − /κ i ϕ i /κ i − ε i +1 − κ i +1 ε i +1 κ i +1 ⊕ I n − − i +1) from equation (16), where χ i ( K ) = κ i , χ i ( F ) = ϕ i /κ i , etc. B is a 2( n − × n −
1) matrix with determinantdet B = κ + ϕ ε κ = f ( χ , χ )where f is the normalization factor of Proposition 6.2. Notice det B is nonzeroexactly when σ i : a → b is admissible. Lemma.
Let A be an N × N matrix. Then up to sign det (cid:16)(cid:94) A (cid:17) = (det A ) N − Proof.
Recall that (cid:86) A = (cid:76) k (cid:86) k A and that (cid:86) k A is the matrix of the k × k minorsof A . Put A in Jordan form and count the appearance of its eigenvalues in each (cid:86) k A to obtain the lemma. (cid:3) Applying the lemma to B ,det (cid:16)(cid:94) B (cid:17) = (det B ) n − , so that det (cid:16)(cid:94) B ⊗ id C (cid:17) = (cid:16) (det B ) n − (cid:17) = (det B ) n − . On the other hand, recall that we normalized F ( σ i ) = (det B ) / · id V ⊗ · · · ⊗ id V i − ⊗ c ⊗ id V i +2 ⊗ · · · ⊗ id V n F ( σ i ) = (det B ) / · id V ∗ ⊗ · · · ⊗ id V ∗ i − ⊗ c ⊗ id V ∗ i +2 ⊗ · · · ⊗ id V ∗ n where the determinants of the matrices on the right-hand side are both 1. It followsthat det (cid:16) F ( σ i ) ⊗ F ( σ i ) (cid:17) = (cid:16) (det B ) / (cid:17) n = (det B ) n − and thus that the normalizations of F (cid:2) F and T agree. The theorem follows. (cid:3) Our main result, proved in §
7, is that T ( L, ρ, ω ) = τ ( L, ρ ), where τ ( L, ρ ) is thetorsion of S \ L twisted by the representation ρ . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 39
Relationship between F and F .Definition 6.9. Recall that for a link L , the mirror L of L is the image of L underan orientation-reversing homeomorphism r : S → S . For an SL ( C )-link ( L, ρ ),the mirror is defined to be (
L, ρ ), where ρ := ρr ∗ is obtained by pulling back from π L to π L along r . Proposition 6.10. ∇ ( L, ρ, ω ) = ∇ ( L, ρ, ω ) . Notice that we want to take the same choice of Casimirs in the mirror.
Proof.
The topological mirroring (
L, ρ ) (cid:55)→ ( L, ρ ) corresponds to a functor M fromˆ B (SL ( C )) ∗ to itself. In terms of braid diagrams, M takes the top-to-bottom mirrorimage and inverts all colors. In more detail, this means that M acts on objects a = ( χ , . . . , χ n ) by M ( χ , . . . , χ n ) = ( χ − n , · · · , χ − )and on braid generators on n strands by M ( σ i ) = σ − n − . We can extend M to ˆ B (SL ( C )) ∗ by leaving the Casimirs unchanged (except forreversing the order.) It is clear that if ( L, ρ, ω ) is the closure of β , then ( L, ρ, ω ) isthe closure of M ( β ). Hence ∇ ( L, ρ, ω ) = t ( F ( M ( β )))On the other hand, up to changing the order of tensor factors, as linear operators F ( M ( β )) = reverse tensor factors F ( β )and because the quantum dimensions are compatible, this implies t ( F ( M ( β ))) = t ( F ( β )) = ∇ ( L, ρ, ω )as claimed. (cid:3) Gauge invariance.Definition 6.11.
Let (
L, ρ ) be a G -link and let g ∈ G . We say that ( L, ρ ) is gauge-equivalent to the link (
L, ρ g ), where ρ g ( x ) = gρ ( x ) g − . Similarly, if ( L, ρ, ω ) is anenhanced SL ( C )-link we say that it is gauge-equivalent to ( L, ρ g , ω ) for g ∈ SL ( C ).The second part is well-defined because the Casimirs are roots of a conjugation-independent quantity. Theorem 6. ∇ is gauge-invariant, i.e. ∇ ( L, ρ, ω ) = ∇ ( L, ρ g , ω ) for any g ∈ SL ( C ) . Similarly ∇ is gauge-invariant.Proof. The theorem for ∇ is a special case of [5, Theorem 5.11], and the same ideaswork for ∇ . The idea is to express the conjugation of SL ( C )-braids in terms ofcertain gauge transformations internal to B (SL ( C )) ∗ . (cid:3) As a corollary, T is gauge-invariant. We will also prove this directly by showingit agrees with the torsion. Schur-Weyl duality
Suppose a = ( χ , . . . , χ n ) is a nonsingular tuple of colors in B ∗ (SL ( C )). We areinterested in the braid group action on tensor products of the form X = X ⊗ · · · ⊗ X n where X i is a U -module with character χ i . To do this, we first want to find thedecomposition of X into irreducible U -modules. One way to do this is by Schur-Weyl duality : the decomposition of X is determined by the subalgebra of End C ( X )commuting with the action of U . In this section, we show that this decompositionis closely related to the Burau representation.The module X has character χ · · · χ n . It is possible for χ · · · χ n to be a singularcharacter even when the χ i are nonsingular. Definition 7.1.
Let a = ( χ , . . . , χ n ) be a tuple of nonsingular colors in B (SL ( C )) ∗ .We say a has nonsingular total holonomy if χ · · · χ n is nonsingular, that is iftr( ψ ( χ · · · χ n )) (cid:54) = ± . The invariants ∇ and T are still well-defined even when a has singular totalholonomy, but the analysis of the tensor decomposition is slightly more complicatedand we can no longer use (2) of Proposition 3.6. For simplicity, we will only consider a with nonsingular total holonomy.Since X is also a module over U /I χ ⊗ · · · ⊗ U /I χ n , finding subalgebras of End C ( X ) commuting with U is closely related to findingsubalgebras of U /I χ ⊗ · · · ⊗ U /I χ n commuting with the image of U under of theiterated coproduct.We show how to find a certain subalgebra C (cid:48) n (super)commuting with the imageof U . The algebra C (cid:48) n is a (nondegenerate) Clifford algebra generated by a 2( n − H (cid:48) n . The braid group action on H (cid:48) n (via the automorphismˇ R ) agrees with the twisted Burau representation (6), so we identify H (cid:48) n with thesubspace of the SL ( C )-twisted cohomology H ( D n ; ρ ) of the n -punctured disc onwhich the braid group acts nontrivially.For simplicity, we first discuss this computation of C (cid:48) n in terms of U and U ⊗ n .There are parallel results for an opposite version C (cid:48) n and the opposite coproducton U . Later we will see how to combine these to the decomposition of the U (cid:2) U -modules in the image of the doubled functor T = F (cid:2) F , and it is this decompositionthat will give the relation with the torsion.7.1. Graded multiplicity spaces for D .Definition 7.2. A superalgebra is a Z / | x | for the degree of x . The supercommutator of x and y is[ x, y ] := xy − ( − | x || y | yx. A super vector space is similarly a Z / f of supervector spaces preserves the grading, and we define the supertrace to bestr f := tr f + − tr f − where f + , f − are the even and odd components of f , respectively. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 41 If W is an ordinary vector space, then the exterior algebra (cid:86) W becomes a supervector space by setting the image of W in (cid:86) W to be odd.Supertraces are related to determinants as follows: Proposition 7.3.
Let W be a vector space of dimension N and A : W → W alinear map. Write (cid:86) A for the induced map (cid:86) W → (cid:86) W on the exterior algebraof W . Then str (cid:16)(cid:94) A (cid:17) = ( − N det(1 − A ) . Proof.
Recall that det( λ − A ) = N (cid:88) k =0 λ N − k ( − N − k tr (cid:18)(cid:94) k A (cid:19) so in particulardet(1 − A ) = ( − N N (cid:88) k =0 ( − k tr (cid:18)(cid:94) k A (cid:19) = ( − N str (cid:16)(cid:94) A (cid:17) . (cid:3) Recall ˜ F = iKF . We can regard U as the algebra generated by K ± , E, ˜ F withrelations EK + KE = ˜ F K + K ˜ F = 0 , E ˜ F − ˜ F E = i K Ωwhere Ω = K − K − (1 + E ˜ F )The choice that E and iKF (instead of KE and F ) are even is for compatibilitywith the map ˇ R . Proposition 7.4. U is a superalgebra with grading | E | = | ˜ F | = 0 , | K | = | Ω | = 1 . The nonsingular representations of § § a = ( ˆ χ , · · · , ˆ χ n ) be a tuple of nonsingular Z -characters with nonsingulartotal holonomy. We want to understand the multiplicity spaces of the T ( a ) as supervector spaces. By Proposition 4.4, F ( a ) ∼ = n − (cid:77) k =1 V ( χ, ω ) ⊕ n − (cid:77) k =1 V ( χ, − ω )where χ = χ · · · χ n is the total holonomy (here χ i = ˆ χ i | Z ) and ω is some arbitrarychoice of Casimir for χ . Similarly F ( a ) ∗ ∼ = n − (cid:77) k =1 V ( χ, ω ) ∗ ⊕ n − (cid:77) k =1 V ( χ, − ω ) ∗ For ε , ε ∈ Z /
2, set W ε ε ( a, ω ) := V ( χ, ( − ε ω ) (cid:2) V ( χ, ( − ε ω ) Each W ε ε is a simple object of D of degree χ (in fact, they are all the simpleobjects of that degree) and we have(11) T ( a ) ∼ = (cid:77) ε ,ε ∈ Z / W ε ε ( a, ω ) ⊗ X ε ε ( a, ω )where the X ε ε ( a, ω ) are multiplicity spaces. Proposition 7.5.
Any endomorphism f ∈ End D ( T ( a )) factors through the decom-position (11) as a direct sum f = (cid:77) ε ,ε ∈ Z / id W ε ε ⊗ g ε ε of endomorphisms g ε ε ∈ End C ( X ε ε ) . In particular, t ( f ) = 1 ω (cid:88) ε ,ε ∈ Z / ( − ε + ε tr g ε ε where t is the modified trace for D .Proof. Recall that the modified dimension of V ( χ, ω ) (cid:2) V ( χ, ω )is ω − . More generally, it is not hard to see from the modified dimensions for C and C that d ( V ( χ, ( − ε ω ) (cid:2) V ( χ, ( − ε ω )) = ( − ε + ε ω (cid:3) We can think of the direct sum of the X ε ε ( a, ω ) as a Z / × Z / ε + ε .We can achieve this flattening by considering the U (cid:2) C U -module W ε ε as a U -module via the coproduct. Then as U -modules, by Proposition 4.7 we have anisomorphism T ( a ) ∼ = (cid:77) ε ∈ Z / P ε ⊗ Y ε ( a )Here by P we mean P and by P we mean P − , the modules of § Y ε ( a )are multiplicity spaces, which in terms of the X ε ε ( a, ω ) are Y ( a ) = X ( a, ω ) ⊕ X ( a, ω ) , Y ( a ) = X ( a, ω ) ⊕ X ( a, ω )We have dropped ω from the notation because (up to change of basis) the decom-position no longer depends on the choice of ω versus − ω . Definition 7.6.
The graded multiplicity space of T ( a ) is the Z / Y ( a ) = Y ( a ) ⊕ Y ( a )with even and odd parts Y ( a ) and Y ( a ), respectively. Corollary 7.7.
Let f ∈ End D ( T ( a )) be an endomorphism. Then f factors througha super vector space endomorphism g of Y ( a ) , and t ( f ) = str gω is up to normalization the supertrace of g . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 43
Burau representations and elements of U ⊗ n . We describe a family ofelements of (a localization of) U ⊗ n related to the Burau representation via thebraiding automorphisms ˇ R and ˇ R ∗ . On a first reading, it is probably more enlight-ening to skip ahead and read the next section, then return here to understand theconstruction.For any element X ∈ U [Ω − ], set X i = 1 ⊗ · · · ⊗ X ⊗ · · · ⊗ ∈ U [Ω − ] ⊗ n with the X in the i th factor. Write { X, Y } = XY + Y X for the anticommutator.
Definition 7.8.
For j = 1 , . . . , n , set α j := K · · · K j − E j Ω − j (12) α j := K · · · K j − ˜ F j Ω − j (13) α j := E j Ω − j K j +1 · · · K n (14) α j := ˜ F j Ω − j K j +1 · · · K n (15)It is not hard to compute that { α j , α k } = 2 δ jk K · · · K j − E j Ω − j { α j , α k } = 2 δ jk K · · · K j − K j F j Ω − j { α j , α k } = 2 iδ jk K · · · K j − (1 − K − j )Ω − j Since Ω ∈ Z , we see that η ( X, Y ) := 12 { X, Y } defines a Z [Ω − ] ⊗ n -valued bilinear form on the Z ⊗ n -module with basis { α νk } .Similarly, we have { α j , α k } = 2 δ jk E j Ω − j K j +1 · · · K n { α j , α k } = 2 δ jk K j F j Ω − j K j +1 · · · K n { α j , α k } = 2 iδ jk (1 − K − j )Ω − j K j +1 · · · K n and there is an analogous form η for the Z ⊗ n -module spanned by the { α νk } . Definition 7.9.
Write H n for the Z ⊗ n -module and C n for Z ⊗ n -algebra generatedby the α νk . C n is the Clifford algebra corresponding to the form η . For k =1 , . . . , n −
1, set β νk := α νk − α νk +1 and write C (cid:48) n ⊂ C n and H (cid:48) n ⊂ H n for the submodule and subalgebra generated bythe operators for k = 1 , . . . , n − ν = 1 ,
2. Similarly write H n , H (cid:48) n , C n , and C (cid:48) n forthe analogous algebras defined in terms of the operators α νk and β νk .Recall that (similar to exterior algebras) Clifford algebras such as C n can bemade into noncommutative superalgebras by choosing the generators to be odd. Proposition 7.10. U supercommutes with C (cid:48) n . That is, for k = 1 , . . . , n − , ν = 1 , , we have { ∆ K, β νk } = 0[∆ E, β νk ] = 0[∆ ˜ F , β νk ] = 0 { ∆Ω , β νk } = 0 Similarly, U cop supercommutes with C (cid:48) n : { ∆ op K, β νk } = 0[∆ op E, β νk ] = 0[∆ op ˜ F , β νk ] = 0 { ∆ op Ω , β νk } = 0 Proof.
The first three lines are straightforward computations using the anticommu-tation relations for the α νk , while the fourth follows from Ω = K − K − (1+ E ˜ F ). (cid:3) Lemma 7.11.
The braiding automorphism acts by ˇ R ( α ) = α ˇ R ( α ) = K α + (1 − K ) α − E ( α − α )ˇ R ( α ) = (1 − K − ) α + K − α + F ( α − α )ˇ R ( α ) = α and similarly the opposite inverse braiding acts by ˇ R ∗ ( α ) = α ˇ R ∗ ( α ) = K − α + (1 − K − ) α + K − E ( α − α )ˇ R ∗ ( α ) = ( K − α + K α + K F ( α − α )ˇ R ∗ ( α ) = α Proof.
Computation. In this context, it is easiest to consider U as an algebra over Z with basis 1 , K, E, ˜ F = iKF . (cid:3) Proposition 7.12.
Write ˇ R i for the action of ˇ R on the i th and i + 1 th tensorfactors. The matrix of ˇ R i acting on H (cid:48) n is (16) I i − ⊕ K − i − F i − K − i F i − E i +1 − K i +1 E i +1 K i +1 ⊕ I n − − i +1) with the matrix action given by right multiplication on row vectors with respect tothe basis { β , β , · · · , β n − , β n − } of H (cid:48) n . Similarly, the matrix of ˇ R ∗ i acting on H (cid:48) n OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 45 is (17) I i − ⊕ K i − K i F i − K i K i F i − K − i +1 E i +1 − K − i +1 K − i +1 E i +1 K − i +1 ⊕ I n − − i +1) These matrices are closely related to the reduced twisted Burau representation(7).
Proof.
This follows immediately from Lemma 7.11. (cid:3)
Schur-Weyl duality for D .Definition 7.13. Set γ νk := α νk (cid:2) K (cid:2) α νk θ νk := β νk (cid:2) K (cid:2) β νk = γ νk − γ νk +1 Write H n for the ( Z (cid:2) Z ) ⊗ n -span of the γ νk and C n for the algebra generatedby H n . Similarly, let H (cid:48) n be the span of the θ νk and C (cid:48) n the algebra generated by H (cid:48) n .Here by X (cid:2) Y we mean an element of U (cid:2) U = U ⊗ C U (the algebra underlying D ) and we write expressions like X (cid:2) Y ⊗ Z (cid:2) W for operators in ( U (cid:2) U ) ⊗ to emphasize the different tensor factors. The multipletensor products here can be hard to parse, so we give an example. If α = X ⊗ X , α = Y ⊗ Y , by α (cid:2) K (cid:2) α we mean( X (cid:2) ⊗ ( X (cid:2)
1) + ( K (cid:2) Y ) ⊗ ( K (cid:2) Y ) . T ( a ) is a U (cid:2) U -module via( X (cid:2) Y ) (cid:55)→ ∆( X ) (cid:2) ∆ op ( Y )but we can use the coproduct U → U (cid:2) U one more time to make it a U -module.For example, E acts by ∆ E (cid:2) ∆ op K + 1 (cid:2) ∆ op E Proposition 7.14.
Let a = ( ˆ χ , · · · , ˆ χ n ) be a tuple of nonsingular Z -characterswith nonsingular total holonomy. Write π a for the structure map π a : ( U (cid:2) U ) ⊗ n → End C ( T ( a )) Then (1) π a ( C n ) is an exterior algebra on n generators, (2) C n acts faithfully on T ( a ) , and (3) thinking of T ( a ) as a U -module, π a ( C (cid:48) n ) super-commutes with U . Proof. (1) We show that the anticommutators π a ( { γ µj , γ νk } )vanish, so that the image is an exterior algebra on the 2 n independent generators π a ( γ νk ), k = 1 , . . . , n , ν = 1 , { α νk , ∆ K } = 0, { γ µj , γ νk } = { α µj , α νk } (cid:2) K (cid:2) { α µj , α νk } By using the anticommutator computations of § j = k . We give the case µ = ν = 1 in detail; the remaining others follow similarly.Observe that { α j , α j } (cid:2) K (cid:2) { α j , α j } = 2 K · · · K j − E j Ω − j (cid:2) K · · · K n (cid:2) E j Ω − j K j +1 · · · K n Write ˆ χ j ( K ) = κ j , ˆ χ j ( E ) = ε j , ˆ χ j (Ω ) = ω j , so that π a ( K j (cid:2)
1) = κ j π a (1 (cid:2) K j ) = κ − j π a ( E j (cid:2)
1) = ε j π a (1 (cid:2) E j ) = − ε j κ − j π a (Ω j (cid:2)
1) = ω j π a (1 (cid:2) Ω j ) = ω j using the fact that the representations in the second half of the (cid:2) product (corre-sponding to F ) use the inverse characters. Hence π a (2 K · · · K j − E j Ω − j (cid:2) K · · · K n (cid:2) E j Ω − j K j +1 · · · K n )= 2 ω j (cid:0) κ · · · κ j − ε j + κ · · · κ n ( − ε j κ − j ) κ − j +1 · · · κ − n (cid:1) = 0as claimed.(2) It is enough to show that the operators π a ( γ νk ) all act independently. Sinceup to a scalar γ k , γ k only act on the k th ⊗ -factor of the product T ( a ) = n (cid:79) j =1 V ( ˆ χ j ) (cid:2) V ( ˆ χ j ) ∗ it is enough to check that γ k and γ k act independently. It is not hard to computeexplicitly that the vectors π a ( γ k ) · v ( ˆ χ , . . . , ˆ χ n ) and π a ( γ k ) · v ( ˆ χ , . . . , ˆ χ n )are independent, where v is the invariant vector of Lemma 6.4, and (2) follows.(3) We can check directly that[∆ E (cid:2) ∆ op K + 1 (cid:2) ∆ op E, θ νk ]= [∆ E (cid:2) ∆ op K + 1 (cid:2) ∆ op E, β νk (cid:2) K (cid:2) β νk ]= [∆ E, β νk ] (cid:2) K (cid:2) [∆ op E, β νk ]= 0using the computations in 7.10. The other generators K, ˜ F follow similarly. (cid:3) Corollary 7.15.
The Z / -graded multiplicity space Y ( a ) of T ( a ) is isomorphic asa vector space to C (cid:48) n . OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 47
Proof.
This is simply an application of (a super version) of the double centralizertheorem. By (3) of Proposition 7.14, π a induces an inclusion C (cid:48) n → Y ( a ), and by(2) this inclusion is injective. But both spaces are dimension 2 n − , so it is anisomorphism. (cid:3) Write (cid:86) B for the (total) exterior power of the twisted reduced Burau represen-tation of Definition 3.3. Theorem 7 (Schur-Weyl duality) . Let a be a nonsingular color in ˆ B (SL ( C )) ∗ with nonsingular total holonomy. Then there exits a family of isomorphisms ϕ a : Y ( a ) → (cid:86) B ( a ) compatible with the braid action in the sense that for any coloredbraid β : a → b , the diagram W ( a ) W ( b ) (cid:86) B ( a ) (cid:86) B ( b ) T ( β ) ϕ a ϕ b (cid:86) B ( β ) commutes.Proof. Let v ( a ) be the invariant vector of Lemma 6.4. By Proposition 7.14 andits corollary, a basis for the multiplicity space Y ( a ) consists of the vectors( θ k · · · θ k s θ k · · · θ k s ) · v ( a )where 1 < k ν · · · < k s ν ≤ n − s ν = 0 , . . . , n − ν = 1 ,
2. It suffices tocheck the action of the braid generators σ i : a → b on the vectors θ νk · v ( a ); if wecan show that this action agrees with the reduced Burau representation (7) thenwe are done.By the definition of the holonomy braiding, we have T ( σ i )( θ νk · v ( a )) = ( ˇ R i,i +1 (cid:2) ˇ R ∗ i,i +1 )( θ νk ) · T ( σ i )( v ( a ))= ( ˇ R i,i +1 (cid:2) ˇ R ∗ i,i +1 )( θ νk ) · v ( b )where we have used the invariance of the family v ( − ) in the second equality. Since( ˇ R i,i +1 (cid:2) ˇ R ∗ i,i +1 )( θ νk ) = ˇ R i,i +1 ( β νk ) (cid:2) ˇ R ∗ i,i +1 (1) + ˇ R i,i +1 (∆ K ) (cid:2) ˇ R ∗ i,i +1 ( β νk )= ˇ R i,i +1 ( β νk ) (cid:2) K (cid:2) ˇ R ∗ i,i +1 ( β νk )we can use the computations of Proposition 7.12.The matrices (16) and (17) have entries in Z ⊗ n , so their action on v ( b ) isdetermined by the characters of b = ( ˆ χ , . . . , ˆ χ n ). Choosing the usual coordinatesˆ χ i ( K i ) = κ i , ˆ χ i ( E i ) = ε i , ˆ χ i ( F i ) = ϕ i /κ i we see that (16) acts by the matrix (7).The opposite braiding ˇ R ∗ corresponds to the matrix (17), and which acts on thesecond (cid:2) factor by the inverse characters ˆ χ − i = ˆ χ i ◦ S . But the antipode S maps(17) to (16), so again (17) acts by (7). (cid:3) Our main result is an immediate corollary.
Theorem 8.
Let ( L, ρ, ω ) be an enhanced SL ( C ) -link. in ˆ B (SL ( C )) ∗ with non-singular holonomy around each strand. Then T ( L, ρ, ω ) = τ ( L, ρ ) is the SL ( C ) -twisted torsion of L . As a consequence, ∇ ( L, ρ, ω ) ∇ ( L, ρ, ω ) = τ ( L, ρ ) so we can interpret ∇ ( L, ρ, ω ) as a nonabelian Conway potential.Proof. By Proposition 3.6 we can represent (
L, ρ ) as the closure of an SL ( C )-braid β with nonsingular total holonomy, and by Proposition 2.10 we can pull β back toa nonsingular SL ( C ) ∗ -braid, possibly after a gauge transformation. Now, by def-inition, T ( L, ρ, ω ) = t ( T ( β )). By Schur-Weyl duality (Theorem 7) the intertwiner T ( β ) factors through the multiplicity superspace Y ( a ) of T ( a ) as (cid:86) B ( β ), so byCorollary 7.7 and Proposition 7.3 we have t ( T ( β )) = str (cid:0)(cid:86) B ( β ) (cid:1) ω = det(1 − B ( β )) ω because the representation B is even-dimensional. Recall that ω = tr( ψ ( a )) − ψ ( a ) is the total holonomy of the color a . But ψ ( a ) ∈ SL ( C ), so det(1 − ψ ( a )) = 2 − tr( ψ ( a )). Hence t ( T ( β )) = det(1 − B ( β )) ω = − det(1 − B ( β ))det(1 − ψ ( a )) = − τ ( L, ρ )by Proposition 3.6. Since τ ( L, ρ ) is only defined up to sign, τ ( L, ρ ) = t ( T ( β )) = T ( L, ρ ) as claimed. The second statement now follows from Proposition 6.10. (cid:3)
Appendix A. Construction of modified traces
We apply the methods of Geer, Kujawa, and Patureau-Mirand [10] to constructthe modified traces of § C -linear category C ,by which mean a pivotal category whose hom spaces are vector spaces over C andwhose tensor product is C -bilinear. C , C , and D (or more generally the category ofrepresentations of a pivotal Hopf C -algebra) are all examples of such categories.More specific results of [10] place extra conditions on C (local finiteness) and oncertain distinguished objects (absolute decomposability, end-nilpotence, etc.) whichare satisfied for finite-dimensional representations of an algebra over an algebraicallyclosed field, perhaps with some diagonalizability assumptions. All our examplessatisfy these hypotheses. That is, for each meridian x , ρ ( x ) does not have 1 as an eigenvalue. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 49
A.1.
Projective objects, ideals, and traces.Definition A.1.
Let C be a category. We say an object P of C is projective if forany epimorphism p : X → Y and any map f : P → Y , there is a lift g : P → X such that the diagram commutes: XP Y pgf
We say I is injective if I is a projective object in C op , i.e. if I satisfies the oppositeof the above diagram. We write Proj( C ) for the class of projective objects of C . Definition A.2.
Let C be a pivotal C -category. A right (left) ideal I is a fullsubcategory of C that is:(1) closed under right (left) tensor products: If V is an object of I and W isany object of C , then V ⊗ W ( W ⊗ V ) is an object of I .(2) closed under retracts: If V is an object of I , W is any object of C , andthere are morphisms f, g with W V W f id W g commuting, then W is an object of I .An ideal of C is a full subcategory which is both a left and right ideal. Proposition A.3.
Let C be a pivotal category. Then the projective and injectiveobjects coincide and Proj( C ) is an ideal.Proof. See [13, Lemma 17]. (cid:3)
Definition A.4.
Let W be an object of a pivotal C -category C . The right partialtrace is the map tr rW : Hom C ( V ⊗ W, X ⊗ W ) → Hom C ( V, X )defined by tr rW ( g ) = (id X ⊗ ev W )( g ⊗ id W ∗ )(id V ⊗ coev W )where ev W : → W ⊗ W ∗ and coev W : W ⊗ W ∗ → are the maps coming fromthe pivotal structure of C and is the tensor unit of C . (See Proposition 5 . I be a right ideal in C . A (right) modified trace (or m-trace) on I is afamily of C -linear functions { t V : Hom C ( V, V ) → C } V ∈ I for every object V of I that are(1) compatible with partial traces: If V ∈ I and W ∈ C , then for any f ∈ Hom C ( V ⊗ W, V ⊗ W ), t V ⊗ W ( f ) = t V (tr rW ( f ))(2) cyclic: If U, V ∈ I , then for any morphisms f : V → U , g : U → V , wehave t V ( gf ) = t U ( f g ) We can similarly define left partial traces and left modified traces. The pivotalstructure on C means that a right modified trace on an ideal will also give a leftmodified trace.With the usual graphical notation for pivotal categories, we can draw the rightpartial trace of a map f : V ⊗ W → X ⊗ W as f XV W Here we are breaking convention by writing the diagram left-to-right instead ofvertically. A.2.
Construction of modified traces.
Let C be a pivotal C -category withtensor unit . Consider the projective cover P → , and assume that P is finite-dimensional. Then P is indecomposable and projective and the space Hom C ( P, )is 1-dimensional over C . Because C is pivotal, P is also injective and Hom C ( , P )is similarly 1-dimensional.The choice of P and a basis of each space are the data necessary to define amodified trace on Proj( C ), which we call a trace tuple. Our definition is a specialcase ([10, § α = β = . Thesemore general traces can be defined for larger ideals than Proj( C ). Definition A.5.
Let C be a pivotal C -category with tensor unit , and let P → be a finite-dimensional cover. ( P, ι, π ) is a trace tuple if P is indecomposable andprojective, ι is a basis of Hom C ( , P ), and π is a basis of Hom C ( P, ). Example A.6.
Let C = C , the category of finite-dimensional U -weight modules,and let P be the projective cover of defined in § V be any of theirreducible 2-dimensional modules of § P , coev V , ev V ) is a trace tuple.Because P is indecomposable, projective, and finite-dimensional, any endomor-phism f ∈ End C ( P ) decomposes f = a + n as an automorphism plus a nilpotentpart. Because C is algebraically closed, a is a scalar, and we write (cid:104) f (cid:105) = a ∈ C .If g ∈ Hom C ( , P ) , h ∈ Hom C ( P, ) are any morphisms, we can similarly define (cid:104) g (cid:105) ι , (cid:104) h (cid:105) π ∈ C by g = (cid:104) g (cid:105) ι ι, h = (cid:104) h (cid:105) π π Lemma A.7.
Let ( P, ι, π ) be a trace tuple. Then for any f ∈ End C ( P ) , (1) πf = (cid:104) f (cid:105) π π (2) f ι = (cid:104) f (cid:105) ι ι (3) (cid:104) f (cid:105) = (cid:104) f ι (cid:105) ι = (cid:104) πf (cid:105) π Proof.
We have f = (cid:104) f (cid:105) id P + n for some nilpotent n . The first statement followsfrom πn = 0. Since π is a basis for Hom C ( P, ), we have πn = λπ for some λ ∈ C .But n k = 0 for some k , so λ k = 0 ⇒ λ = 0 because C is an integral domain. When drawing string diagrams in this manner, we interpret the “right” in right trace tomean “on the right as seen by f .” If C is semisimple, then is projective and we recover the usual trace in a pivotal category. OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 51
The second statement follows from a similar argument, and the third from the firsttwo. (cid:3)
Lemma A.8.
Let ( P, ι, π ) be a trace tuple for C and V a projective object. Thenthere are maps σ V : P ⊗ V → V , τ V : V → P ⊗ V such that the diagrams commute: P ⊗ VV V ∼ = ⊗ V σ V ι ⊗ id V ι ⊗ id V P ⊗ VV V ∼ = ⊗ V π ⊗ id V τ V id V Proof. V is projective and π ⊗ id V : P ⊗ V → ⊗ V → V is an epimorphism, so alift τ V exits. The dual argument works for σ V . (cid:3) Theorem 9.
Let ( P, ι, π ) be a trace tuple for C and choose maps as in Lemma A.8.Then there exits a right modified trace on Proj( C ) defined for f ∈ Hom C ( V, V ) by t V ( f ) = (cid:104) tr rV ( τ V f ) (cid:105) ι = (cid:104) tr rV ( σ V f ) (cid:105) π This is a special case of [10, Theorem 4.4].
Proof.
In the diagrams in this proof, we identifyEnd C ( P ) /J ∼ = Hom C ( , P ) ∼ = Hom C ( P, ) ∼ = C via the maps (cid:104)−(cid:105) , (cid:104)−(cid:105) ι , and (cid:104)−(cid:105) π . Here J is the ideal of nilpotent elements ofEnd C ( P ), so when we draw a diagram representing a morphism P → P we reallymean its image in this quotient. τ V and σ V exist by Lemma A.8, but are not unique. We show that the tracedoes not depend on the choice of either. In graphical notation, tr rV ( τ V f ) can bewritten as f τ V P Since σ V ( ι ⊗ id V ) = id V , we can rewrite this morphism as f τ V Pσ V ι where ι has no left-hand arrows because it is a map → P . By Lemma A.7, theabove diagram is equal to f τ V P σ V π But since ( π ⊗ id V ) τ V = id V , this is equal to tr rV ( f σ V ): fP σ V It follows that (cid:104) tr rV ( τ V f ) (cid:105) ι = (cid:104) tr rV ( σ V f ) (cid:105) π as claimed.To check the compatibility with the partial trace, let f : V ⊗ W → V ⊗ W .Choose τ V with ( π ⊗ id V ) τ V = id V , and notice that we can set τ V ⊗ W = τ V ⊗ id W .Then t V ⊗ W ( f ) is f τ V PWV which is clearly equal to t V (tr rW ( f )).Finally, we show cyclicity. Suppose f : V → W and g : W → V . Then t V ( gf ) isequal to fσ V g fσ V g = by the cyclicity of the usual trace. But by inserting ( π ⊗ id W ) τ W = id W and thenapplying Lemma A.7 as before, we can rewrite this as OLONOMY INVARIANTS OF LINKS AND NONABELIAN REIDEMEISTER TORSION 53 fσ V g τ W πfσ V g τ W ι = By absorbing ι into σ V , we see that this is equal to fg τ W = t W ( f g ) . (cid:3) It can be shown that the modified trace on Proj( C ) is essentially unique; choosingdifferent ι or π will simply change t by an overall scalar. The paper [10] provesthis and a number of other useful results about these modified traces, such asnon-degeneracy and compatibility with the left-hand version of the construction.A.3. Application to U . Recall the projective U -module P defined in § x, y , y , z . As before, we can describe the action of E and F via the diagram xy y z E FF E
Write ι : → P for the linear map sending 1 ∈ C to z , and π : P → for theprojection onto the subspace spanned by x . It is not hard to see that these aremorphisms of U -modules. Proposition A.9. ( P , ι, π ) is a trace tuple defining the modified trace of Theorem2.Proof. It is clear from Proposition 4.7 that it is a trace tuple, so it suffices to checkthat it gives the same renormalized dimensions as in Theorem 2. Let V = V ( χ, ω )be an irreducible 2-dimensional module. It is not difficult to find a U -module map τ V with P ⊗ VV V ∼ = ⊗ V π ⊗ id V τ V id V Then we can check that tr rV ( τ v id V ) = 2 ω ι and since we chose 2 ι in our trace tuple the renormalized dimension of V is ω − asclaimed. (cid:3) We conclude by the constructing the modified traces for D . Proof of Theorem 3.
The modified trace on D is constructed using the trace tuple( P (cid:2) P , ι (cid:2) ι, π (cid:2) π )obtained as the product of the tuples for C and C . (Recall that P ∗ ∼ = P .) We showthat this trace is compatible with the traces on the factors, in the sense that if V, V are objects and f : V → V , g : V → V are morphisms in C and C , respectively,then t ( f (cid:2) g ) = t ( f ) t ( g )The computation of the renormalized dimensions for D follows immediately.Choose lifts τ V , τ V as usual. Then the diagram( P (cid:2) P ) ⊗ ( V (cid:2) V ) V (cid:2) V V (cid:2) V ( π (cid:2) π ) ⊗ (id V (cid:2) id V ) τ V (cid:2) τ V id V (cid:2) V commutes, so τ V (cid:2) τ V is a lift for V (cid:2) V . But then we can use the compatibility ofthe pivotal structures to write t ( f (cid:2) g ) = (cid:10) tr rV (cid:2) V (( τ V (cid:2) τ V )( f (cid:2) g )) (cid:11) ι (cid:2) ι = (cid:10) tr rV ( τ V f ) (cid:2) tr rV ( τ V g ) (cid:11) ι (cid:2) ι = (cid:104) tr rV ( τ V f ) (cid:105) ι (cid:10) tr rV ( τ V g ) (cid:11) ι = t ( f ) t ( g ) . (cid:3) References [1] Benjamin Balsam.
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