Link invariants from finite categorical groups
aa r X i v : . [ m a t h . G T ] D ec Link invariants from finite categorical groups
Jo˜ao Faria Martins and Roger Picken Departamento de Matem´atica, Faculdade de Ciˆencias e Tecnologia, Universidade Novade Lisboa, Quinta da Torre, 2829-516 Caparica. Portugal. [email protected] address:
Department of Pure Mathematics, School of Mathematics, Universityof Leeds, Leeds, LS2 9JT, UK. Center for Mathematical Analysis, Geometry and Dynamical Systems, MathematicsDepartment, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais,1049-001 Lisboa Portugal. [email protected]
October 27, 2018
Abstract
We define an invariant of tangles and framed tangles given a finite crossed module and a pairof functions, called a Reidemeister pair, satisfying natural properties. We give several examples ofReidemeister pairs derived from racks, quandles, rack and quandle cocycles, and central extensionsof groups. We prove that our construction includes all rack and quandle cohomology (framed) linkinvariants, as well as the Eisermann invariant of knots. We construct a class of Reidemeister pairswhich constitute a lifting of the Eisermann invariant, and show through an example that this classis strictly stronger than the Eisermann invariant itself.
Key words and phrases:
Knot invariant, tangle, peripheral system, quandle, rack, crossed mod-ule, categorical group, non-abelian tensor product of groups.
In knot theory, for a knot K , the fundamental group π ( C K ) of the knot complement C K , also knownas the knot group, is an important invariant, which however depends only on the homotopy typeof the complement of K (for which it is a complete invariant), and therefore, for example, it fails todistinguish between the square knot and the granny knot, which have homotopic, but non-diffeomorphiccomplements. Nevertheless, a powerful knot invariant I G can be defined from any finite group G , bycounting the number of morphisms from the knot group into G . In a recent advance, Eisermann [13]constructed, from any finite group G and any x ∈ G , an invariant E ( K ) that is closely associatedto a complete invariant [29], known as the peripheral system, consisting of the knot group π ( C K )and the homotopy classes of a meridian m and a longitude l . Eisermann gives examples showing thathis invariant is capable of distinguishing mutant knots as well as detecting chiral (non-obversible),non-inversible and non-reversible knots (using the terminology for knot symmetries employed in [13]).Explicitly the Eisermann invariant for a knot K is: E ( K ) = X { f : π ( C K ) → G | f ( m )= x } f ( l ) , and takes values in the group algebra Z [ G ] of G . 1isermann’s invariant has in common with many other invariants that it can be calculated bysumming over all the different ways of colouring knot diagrams with algebraic data. Another well-knownexample of such an invariant is the invariant I G defined above, which can be calculated by countingthe number of colourings of the arcs of a diagram with elements of the group G , subject to certain(Wirtinger) [8] relations at each crossing. Another familiar construction is to use elements of a finitequandle to colour the arcs of a diagram, satisfying suitable rules at each crossing [17, 10]. Note that thefundamental quandle of the knot complement is a powerful invariant that distinguishes all knots, up tosimultaneous orientation reversal of S and of the knot (knot inversion); see [19]. Other invariants refinethe notion of colouring diagrams by assigning additional algebraic data to the crossings - a significantexample is quandle cohomology [10]. Eisermann’s invariant can be viewed in several different ways, butfor our purpose the most useful way is to see it as a quandle colouring invariant using a special quandle(the “Eisermann quandle”) associated topologically with the longitude and so-called partial longitudescoming from the diagram.A diagram D of a knot or link K naturally gives rise to a particular presentation of the knot group,known as the Wirtinger presentation. Our first observation is that there is also a natural crossed moduleof groups associated to a knot diagram [4, 2, 15], namely Π ( X D , Y D ) = (cid:0) ∂ : π ( X D , Y D ) → π ( Y D ) (cid:1) -see the next section for the definition of a crossed module of groups and the description of Π ( X D , Y D ).This crossed module is a totally free crossed module [4], where π ( Y D ) is the free group on the arcs of D and Π ( X D , Y D ) is the free crossed module on the crossings of D . The crossed module Π ( X D , Y D ) is notitself a knot invariant, although it can be related to the knot group since π ( C K ) = coker( ∂ ). However,up to crossed module homotopy [4], Π ( X D , Y D ) is a knot invariant, depending only on the homotopytype of the complement C K . Therefore, given a finite crossed module G = ( ∂ : E → G ), one can definea knot invariant I G by counting all possible colourings of the arcs and crossings of a diagram D withelements of G and E respectively, satisfying some natural compatibility relations (so that colouringscorrespond to crossed module morphisms Π ( X D , Y D ) → G ), and then normalising [14, 15].This invariant I G ( K ) depends only on the homotopy type of the complement C K [15, 16], thus itis a function of the knot group alone. Our main insight is that imposing a suitable restriction on thetype of such colourings, and then counting the possibilities, gives a finer invariant. The restriction isto colour the arcs and crossings in a manner that is a) compatible with the crossed module structure,and b) such that the assignment at each crossing is given in terms of the assignments to two incomingarcs by two functions (one for each type of crossing), termed a Reidemeister pair. Since we are choosingparticular free generators of Π ( X D , Y D ) this takes away the homotopy invariance of the invariant.The two functions making up the Reidemeister pair must satisfy some conditions, and dependingon the conditions imposed, our main theorem (Theorem 3.19) states that one obtains in this way aninvariant either of knots or of framed knots (knotted ribbons). In fact our statement extends to tanglesand framed tangles.This invariant turns out to have rich properties, which are described in the remainder of the paper(Section 4). It includes as special cases the invariants coming from rack and quandle colourings, fromrack and quandle cohomology and the Eisermann invariant (subsections 4.1. and 4.2). In section 4.3we introduce the notion of an Eisermann lifting, namely a Reidemeister pair derived from a centralextension of groups which reproduces the arc colourings of the Eisermann quandle, combined withadditional information on the crossings. We give a simple example of an Eisermann lifting that isstrictly stronger than the Eisermann invariant it comes from. Finally, in subsection 4.4, we give ahomotopy interpretation of the Eisermann liftings, by using the notion of non-abelian tensor productand non-abelian wedge product of groups, defined by Brown and Loday [6, 7]. Definition 2.1 (Crossed module of groups)
A crossed module of groups, G = ( ∂ : E → G, ⊲ ) , isgiven by a group morphism ∂ : E → G together with a left action ⊲ of G on E by automorphisms, such g g g ggge∂ ( e ) Figure 1: The action of an element g ∈ π ( Y ) on an e ∈ π ( X, Y ). that the following conditions (called Peiffer equations) hold:1. ∂ ( g ⊲ e ) = g∂ ( e ) g − ; ∀ g ∈ G, ∀ e ∈ E, ∂ ( e ) ⊲ f = ef e − ; ∀ e, f ∈ E .The crossed module of groups is said to be finite, if both groups G and E are finite. Morphisms ofcrossed modules are defined in the obvious way. Example 2.2
Any pair of finite groups G and E , with E abelian, gives a finite crossed module of groupswith trivial ∂, ⊲ (i.e. ∂ ( E ) = 1 , g ⊲ e = e, ∀ g ∈ G, ∀ e ∈ E ). More generally we can choose any action of G on E by automorphisms, with trivial boundary map ∂ : E → G . Example 2.3
Let G be any finite group. Let Ad denote the adjoint action of G on G . Then (id : G → G, Ad) is a finite crossed module of groups.
Example 2.4
There is a well-known construction of a crossed module of groups in algebraic topology,namely the fundamental crossed module associated to a pointed pair ( X, Y ) of path-connected topologicalspaces ( X, Y ) , thus Y ⊂ X , namely the crossed module: Π ( X, Y ) = ( ∂ : π ( X, Y ) → π ( Y ) , ⊲ ) , withthe obvious boundary map ∂ : π ( X, Y ) → π ( Y ) , and the usual action of π ( Y ) on π ( X, Y ) ; see figure1, and [4] for a complete definition. This is an old result of Whitehead [30, 31]. Example 2.5
We may construct a topological pair ( X D , Y D ) from a link diagram D of a link K in S ,and thus obtain a crossed module Π ( X D , Y D ) , associated to the diagram. Regard the diagram as theorthogonal projection onto the z = 0 plane in S = R ∪ {∞} of a link K D , isotopic to K , lying entirelyin the plane z = 1 , except in the vicinity of each crossing point, where the undercrossing part of the linkdescends to height z = − . Then we take X D (an excised link complement) to be the link complement C K of K D minus an open ball, and Y D to be the z ≥ subset of X D , i.e. X D := (cid:0) S \ n ( K D )) ∩{ ( x, y, z ) | z ≥ − } , Y D := X D ∩ { ( x, y, z ) | z ≥ } , where n ( K D ) is an open regular neighbourhood of K D in S . Note that the space X D depends only on K itself, so we can write it as X K . The same is not true for Y D .Each arc of the diagram D corresponds to a generator of π ( Y D ) and there are no relations betweengenerators. Each crossing of the diagram D corresponds to a generator of π ( X D , Y D ) , namely ǫ :[0 , → X D , where the image under ǫ of the interior of [0 , lies entirely in the region z < andthe image of the boundary of [0 , , a loop contained in Y D , encircles the crossing as in Figure 2. Thisboundary loop is the product of four arc loops in π ( Y D ) . Again there are no (crossed module) relationsbetween the generators of π ( X D , Y D ) associated to the crossings. (This can be justified by Whitehead’stheorem [30, 31, 4]: for path-connected spaces X, Y , if X is obtained from Y by attaching 2-cells, then Π ( X, Y ) is the free crossed module on the attaching maps of the 2-cells. Note that X D is homotopyequivalent to the CW-complex obtained from Y D by attaching a 2-cell for each crossing).We observe that the quotient π ( Y D ) / im ∂ is isomorphic to the fundamental group of the link comple-ment C K = π ( S \ n ( K )) for any diagram D , since quotienting π ( Y D ) by im ∂ corresponds to imposingthe Wirtinger relations [8], which produces the Wirtinger presentation of π ( C K ) , coming from the par-ticular choice of diagram. Thus Π ( X D , Y D ) , whilst not being itself a link invariant (unless [15, 16]considered up to crossed module homotopy), contains an important link invariant, namely π ( C K ) , bytaking the above quotient. The guiding principle in the construction to follow is to extract additionalReidemeister invariant information from the crossed module Π ( X D , Y D ) . π ( Y D ) and a generator of π ( X D , Y D ). C ( G ) defined from a categorical group G It is well-known that a crossed module of groups G gives rise to a categorical group, denoted C ( G ),a monoidal groupoid where all objects and arrows are invertible, with respect to the tensor productoperation; see [2, 4, 1, 14, 21]. We recall the essential details. Given a crossed module of groups G = ( ∂ : E → G, ⊲ ), the monoidal category C ( G ) has G as its set of objects, and the morphisms from U ∈ G to V ∈ G are given by all pairs ( U, e ) with e ∈ E such that ∂ ( e ) = V U − . It is convenient tothink of these morphisms as downward pointing arrows and / or to represent them as squares - see (1). U ( U,e ) (cid:15) (cid:15) V or UeV , with ∂ ( e ) U = V. (1)The composition of morphisms U e → V and V f → W is defined to be U fe −→ W , and the monoidalstructure ⊗ is expressed as U ⊗ V = U V on objects, and, on morphisms, as: U ( U,e ) y V ⊗ W ( W,f ) y X = U W y(cid:0)
UW, ( V ⊲f ) e (cid:1) W X .
These algebraic operations are shown using squares in (2) and (3).
UeVfW = UfeW (2)
UeV ⊗ U ′ e ′ V ′ = UU ′ ( V ⊲e ′ ) eV V ′ (3)The (strict) associativity of the composition and of the tensor product are trivial to check. The functo-riality of the tensor product (also known as the interchange law) follows from the 2nd Peiffer equation.This calculation is done for example in [2, 4, 26].Let κ be any commutative ring. The monoidal category C ( G ) has a κ -linear version C κ ( G ), whoseobjects are the same as the objects of C ( G ), but such that the set of morphisms U → V in C κ ( G ) is4igure 3: A tangle with source + + −− and target + − .Figure 4: Elementary generators of tangle diagrams.given by the set of all κ linear combinations of morphisms U → V in C ( G ). The composition and tensorproduct of morphisms in C κ ( G ) are the obvious linear extensions of the ones in C ( G ). It is easy to seethat C κ ( G ) is a monoidal category. The categorical group formalism is very well matched to the categoryof tangles to be used in the next section. G colourings of oriented tangle diagrams Tangles are a simultaneous generalization of braids and links. We follow [24, 20] very closely, to whichwe refer for more details. Recall that an embedding of a manifold T in a manifold M is said to be neatif ∂ ( T ) = T ∩ ∂ ( M ). Definition 3.1
An oriented tangle [20, 28, 24] is a 1-dimensional smooth oriented manifold neatlyembedded in R × R × [ − , , such that ∂ ( T ) ⊂ N × { } × {± } . A framed oriented tangle is a tangletogether with a choice of a framing in each of its components [20, 24]. Alternatively we can see a framedtangle as an embedding of a ribbon into R × R × [ − , , [27, 28, 11]. Definition 3.2
Two oriented tangles (framed oriented tangles) are said to be equivalent if they arerelated by an isotopy of R × R × [0 , , relative to the boundary. Definition 3.3
A tangle diagram is a diagram of a tangle in R × [ − , , obtained from a tangle byprojecting it onto R × { } × [ − , . Any tangle diagram unambiguously gives rise to a tangle, up toequivalence. We have monoidal categories of oriented tangles and of framed oriented tangles [20, 28], wherecomposition is the obvious vertical juxtaposition of tangles and the tensor product T ⊗ T ′ is obtainedby placing T ′ on the right hand side of T . The objects of the categories of oriented tangles and offramed oriented tangles are words in the symbols { + , −} ; see figure 3 for conventions.An oriented tangle diagram is a union of the tangle diagrams of figure 4, with some vertical linesconnecting them. This is a redundant set if we consider oriented tangles up to isotopy. Definition 3.4
A sliced oriented tangle diagram is an oriented tangle diagram, subdivided into thinhorizontal strips, inside which we have only vertical lines and possibly one of the morphisms in figure 5.
A theorem appearing in [20] (Theorem XII.2.2) and also in [18, 27, 28, 24] states that the categoryof oriented tangles may be presented in terms of generators and relations as follows:5 + X − ∪ ← ∪ ∩ ← ∩ Figure 5: Generators for the categories of oriented tangles and of framed oriented tangles. ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = R A R BR C R DR R ′ R A R R B R C Figure 6: Relations for the categories of oriented tangles and of framed oriented tangles.
Theorem 3.5
The monoidal category of oriented tangles is equivalent to the monoidal category pre-sented by the six oriented tangle diagram generators : X + , X − , ∪ , ← ∪ , ∩ , ← ∩ , (4) shown in Figure 5, subject to the 15 tangle diagram relations R A − D , R , R A − C , R of Figure 6.The category of framed oriented tangles has the same set (4) as generators, subject to the 15 relations R A − D , R ′ , R A − C , R of Figure 6. Remark 3.6
We have replaced the R3 relation of Kassel’s theorem with its inverse, since this is slightlymore convenient algebraically. The two forms of R3 are equivalent because of the R2A relations. Thesix other types of oriented crossing in figure 4, with one or both arcs pointing upwards, can be expressedin terms of the generators (4) and are therefore not independent generators - see [20], Lemma XII.3.1.
The previous theorem gives generators and relations at the level of tensor categories. If we wantto express not-necessarily-functorial invariants of tangles it is more useful to work with sliced tanglediagrams. The following appears for example in [24]:
Theorem 3.7
Two sliced oriented tangle diagrams represent the same oriented tangle (framed orientedtangle) if, and only if, they are related by1. Level preserving isotopies of tangle diagrams. rivial tangle diagram trivial tangle diagramtrivial tangle diagram trivial tangle diagram trivial tangle diagramtrivial tangle diagram T TT S T S ∼ = ∼ = Figure 7: The identity move and the interchange move. a b c de f
Figure 8: An X -enhanced tangle with source a.b.c ∗ .d ∗ and target e.f ∗ .
2. The moves R A − R D , R , R A − R C , R of Figure 6 (in the case of tangles), performedlocally in a diagram, or the moves R A − R D , R ′ , R A − R C , R of Figure 6, in the case offramed tangles.3. The “identity” and “interchange” moves of figure 7. (Here T and S can be any tangle diagramsand a trivial tangle diagram is a diagram made only of vertical lines.) Definition 3.8 (Enhanced tangle)
Let X be a set (normally X will be either a group or a quandle/ rack). An X -enhanced (framed) oriented tangle is a (framed) oriented tangle T together with anassignment of an element of X to each point of the boundary of T . We will consider X -enhanced(framed) tangles up to isotopy of R × R × [ − , , fixing the end-points. Given a set X , there exist monoidal categories having as morphisms the set of X -enhanced orientedtangles and of X -enhanced framed oriented tangles, up to isotopy. These categories have as objects theset of all formal words ω in the symbols a and a ∗ where a ∈ X . See figure 8 for conventions. Definition 3.9
Let G be a group. There exists an evaluation map ω e ( ω ) , which associates to aword ω in G ⊔ G ∗ an element of G , obtained by multiplying all elements of ω in the same order, byputting g ∗ . = g − (and e ( ∅ ) = 1 G for the empty word ∅ ). Let G = ( ∂ : E → G, ⊲ ) be a crossed module of groups. We wish to define the notion of a G -colouring ofan oriented tangle diagram, by assigning elements of G to the arcs and elements of E to the crossingsin a suitable way.For a link diagram D realized as a tangle diagram, a G -colouring may be regarded as a morphismof crossed modules from the fundamental crossed module Π ( X D , Y D ) of Example 2.5, to G . This ideaextends in a natural way to general tangle diagrams.7 ●●●●●●●●● X { { ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ e ●●●●●●●●● X Y
ZXeXY X ●●●●●●●●●●●●●●●●●●●● Z { { ✇✇✇✇✇✇✇✇✇ e { { ✇✇✇✇✇✇✇✇✇ Y X
XZeY XX J J X ¯ X E G X T T ¯ XX E G X (cid:20) (cid:20) G E ¯ XX X (cid:10) (cid:10) G E X ¯ XX (cid:15) (cid:15) X E X X O O ¯ X E ¯ X Figure 9: Turning G -coloured tangles into morphisms of C ( G ). The symbol ¯ X stands for X − . Definition 3.10
Given a finite crossed module G = ( ∂ : E → G, ⊲ ) and an oriented tangle diagram D ,a G -colouring of D is an assignment of an element of G to each arc of the diagram, and of an elementof E to each crossing of the diagram, such that, at each crossing of type X + or X − with colourings asin (7) , the following relations hold: X + : ∂ ( e ) = XY X − Z − (5) X − : ∂ ( e ) = Y XZ − X − (6) Z ●●●●●●●●● X { { ✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇✇ e ●●●●●●●●● X Y X ●●●●●●●●●●●●●●●●●●●● Z { { ✇✇✇✇✇✇✇✇✇ e { { ✇✇✇✇✇✇✇✇✇ Y X (7)Thus we are assigning to each type of coloured crossing a morphism of C ( G ) and of C κ ( G ), and in asimilar way we may associate morphisms of C ( G ) to all elementary G -coloured tangles, as summarised infigure 9. With the duality where the dual of the morphism X e −→ ∂ ( e ) X is X − ∂ ( e ) − X − ⊲e −−−−→ X − , andthe morphisms associated to the cups and caps are the ones in figure 9, we can easily see [14] that themonoidal category C ( G ) is a compact category [27], and in fact a pivotal category [3], which however isnot spherical in general. Therefore planar graphs coloured in C ( G ) can be evaluated to give morphismsin C ( G ), and this evaluation is invariant under planar isotopy.Thus we can assign to the complete G -coloured oriented tangle diagram a morphism of C ( G ), by usingthe monoidal product horizontally and composition vertically. This leads to the following definition: Definition 3.11
Given a G -colouring F of a tangle diagram D (we say F ∈ C G ( D ) , the set of G colourings of D ), the evaluation of F , denoted e ( F ) , is the morphism in C ( G ) obtained by multiplyinghorizontally and composing vertically the morphisms of C ( G ) associated to the elementary tangles whichmake up D . emark 3.12 For a link diagram the evaluation of F takes values in A , the automorphism subgroupof G , i.e. A = ker ∂ ⊂ E . For a tangle diagram without open ends at the top and bottom, i.e. a link diagram, one canconjecture that the number of G -colourings of the diagram can be normalized to a link invariant, byanalogy with the familiar link invariant which is the number of Wirtinger colourings of the diagramusing a finite group G (a Wirtinger colouring in the present context would be a G -colouring wherethe group E is trivial). Indeed it was proven in [14] that the number of colourings of a link diagramevaluating to the identity of E can be normalised to give an invariant of knots. However this invariantdepends only on the homotopy type of the complement of the knot [15], thus it is a function of the knotgroup only.Therefore we are led to consider the possibility of imposing more refined constraints on the G -colourings of a tangle diagram in such a way that the number of constrained G -colourings does respectthe Reidemeister moves. Intuitively we are looking at the simple homotopy type, rather than thehomotopy type of a link complement. Our idea is to restrict ourselves to G -colourings of diagramswhere, at each crossing, the colouring of the crossing with an element of E is determined by the G -colouring of two arcs, namely the overcrossing arc and the lower undercrossing arc. To this end weintroduce two functions: ψ : G × G → E, φ : G × G → E, which determine the E -colouring of the two types of crossing, as in (8): Z ❅❅❅❅❅❅❅❅ ψ ( X,Y ) X ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅ X Y X ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ φ ( X,Y ) Z ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ~ ~ ⑦⑦⑦⑦⑦⑦⑦ Y X (8)Since this is a G -colouring, these functions determine the G -assignment for the remaining arc: X + : Z = ∂ψ ( X, Y ) − XY X − (9) X − : Z = X − ∂φ ( X, Y ) − Y X (10)We now come to our main definition.
Definition 3.13 (unframed Reidemeister pair)
The pair
Φ = ( ψ, φ ) is said to be an unframedReidemeister pair if ψ : G × G → E and φ : G × G → E satisfy the following three relations for each X, Y, T ∈ G : ψ ( X, X ) R = 1 E (11) φ ( X, Y ) ψ ( X, Z ) R = 1 E (12) φ ( Y, X ) .Y ⊲ φ ( T, Z ) .φ ( T, Y ) R = X ⊲ φ ( T, Y ) .φ ( T, X ) .T ⊲ φ ( V, W ) (13) where, in R2, Z = X − ∂φ ( X, Y ) − Y X , and in R3: Z = Y − ∂φ ( Y, X ) − XY , V = T − ∂φ ( T, Y ) − Y T , W = T − ∂φ ( T, X ) − XT Remark 3.14
The equations above relate to the Reidemeister 1-3 moves, as we will see shortly in theproof of Theorem 3.19. If equation (12) holds we can substitute (13) by the equivalent: ψ ( X, Y ) . A ⊲ ψ ( X, Z ) . ψ ( A, B ) =
X ⊲ ψ ( Y, Z ) . ψ ( X, C ) . D ⊲ ψ ( X, Y ) (14) where: A = ∂ ( ψ ( X, Y )) − XY X − , B = ∂ ( ψ ( X, Z )) − XZX − ,C = ∂ ( ψ ( Y, Z )) − Y ZY − , D = ∂ ( ψ ( X, C )) − XCX − . f ( Z ) f ( Z ) f ( Z ) φ ( f ( Z ) ,Z ) A AAg ( A ) ψ ( A,A ) Figure 10: Definition of f, g : G → G . Definition 3.15 (Framed Reidemeister pair)
The pair
Φ = ( ψ, φ ) is said to be a framed Reide-meister pair if relations R and R of Definition 3.13 hold and moreover:1. Given Z in G , the equation (c.f. left of figure 10) ∂ ( φ ( A, Z )) A = Z has a unique solution f ( Z ) ∈ G .2. Defining g ( A ) = ∂ ( ψ ( A, A )) − A (c.f. right of figure 10) it holds that f ◦ g = g ◦ f = id . Inparticular both f and g are bijective. Definition 3.16
Given a crossed module G = ( ∂ : E → G, ⊲ ) , with G finite, provided with a (framed orunframed) Reidemeister pair Φ = ( ψ, φ ) , and an oriented G -enhanced tangle diagram D , a Reidemeister G -colouring of D is a G -colouring of D (which extends the colourings at the end-points of D , in the sensethat an arc coloured by g corresponds to endpoints coloured by g or g ∗ , depending on the orientation -see Figure 8) determined by the functions ψ : G × G → E, φ : G × G → E , which fix the colourings ateach crossing as in (8) , (9) and (10) . We are now in a position to define a state-sum coming from the Reidemeister G -colourings of a linkdiagram D . Recall Definitions 3.9 and 3.11. Definition 3.17
Consider a crossed module G = ( ∂ : E → G, ⊲ ) , with G finite, provided with a (framedor unframed) Reidemeister pair Φ = ( ψ, φ ) . Consider an oriented G -enhanced tangle diagram D ,connecting the words ω and ω ′ in G ⊔ G ∗ . We denote the corresponding set of Reidemeister G -colouringsof D by C Φ ( D, ω, ω ′ ) . Then we define the state sum: I Φ ( D ) = h ω | I Φ ( D ) | ω ′ i . = X F ∈ C Φ ( D,ω,ω ′ ) e ( F ) (15) taking values in N (cid:2) Hom C ( G ) (cid:0) e ( ω ) , e ( ω ′ ) (cid:1)(cid:3) ⊂ Hom C Z ( G ) (cid:0) e ( ω ) , e ( ω ′ ) (cid:1) . (Here the set of morphisms x → y in a category C is denoted by Hom C ( x, y ) .) Remark 3.18 If D is a link diagram then I Φ ( D ) takes values in Z [ A ] , the group algebra of A = ker ∂. Theorem 3.19
The state sum I Φ defines an invariant of G -enhanced tangles if Φ is an unframedReidemeister pair and an invariant of framed G -enhanced tangles if Φ is a framed Reidemeister pair. Proof.
To prove this result, we need to show that I Φ respects the relations of Theorem 3.7. Invarianceunder level preserving isotopy is obvious. Let us now address, for the unframed case, the moves R A − D , R R A − C , R C ( G ) are equal(and in some cases, that the remaining arcs connecting to the exterior are also coloured compatibly).Thus for a pair of diagrams related by one of the relations, each term in the expression for I Φ for onediagram has a corresponding term in the expression for I Φ for the other diagram, and the evaluationsare equal term by term.R0A and R0B: Fix X ∈ G . The corresponding equation in E is 1 E = 1 E in each case.10 = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = R A R BR C R DR R ′ R A R R B R CX X X X X XY XX Z Y X X Z Z X X Y Z XX YWX X XY XY Z Z W VZW XY ZX Y X Y UTY X T V UY ZX Y T T V ZWX Y TX Y ZX W X Y U Y TX Y X WY X Z X Z X ZTU Z
Figure 11: Tangle relations with G -assignments to the arcs.R0C: Fix X, Y ∈ G . The corresponding equation in E :( X − Z − ) ⊲ ψ ( X, Y ) = ( Y − X − ) ⊲ ψ ( X, Y )is an identity which follows from: ψ ( X, Y ) = ( ∂ψ ( X, Y )) ⊲ ψ ( X, Y ) = (
XY X − Z − ) ⊲ ψ ( X, Y ) . R0D: Fix
X, Y ∈ G . The corresponding equation in E :( Z − X − ) ⊲ φ ( X, Y ) = ( X − Y − ) ⊲ φ ( X, Y )is an identity which follows from: φ ( X, Y ) = ( ∂φ ( X, Y )) ⊲ φ ( X, Y ) = (
Y XZ − X − ) ⊲ φ ( X, Y ) . R2A: Fix
X, Y ∈ G . The corresponding equation in E is: φ ( X, Y ) ψ ( X, Z ) = 1 E = ψ ( Y, X ) φ ( Y, T ) (16)where Z = X − ∂φ ( X, Y ) − Y X and T = ∂ψ ( Y, X ) − Y XY − . The 1st equality is (12), and the 2ndequality follows from the 1st: ψ ( X, Z ) φ ( X, Y ) = 1 E with Z = X − ∂φ ( Y, X ) − XY , i.e. Y = ∂φ ( X, Y ) XZX − = ∂ψ ( X, Z ) − XZX − , then substitute variables X Y, Z X, Y T . Applying ∂ to (16), it follows that W = Y and U = X .R1: Fix X ∈ G . The corresponding equation in E is: ψ ( X, X ) = 1 E = φ ( X, Y ) , ∂φ ( X, Y ) =
Y X − W = X , and the 2nd equality follows from (11) and (16), see justbelow, which also implies Y = X . φ ( X, Y ) = φ ( X, Y ) ψ ( X, X ) φ ( X, X ) = φ ( X, X ) = 1 E . R2B: Fix
X, Y ∈ G . The corresponding equation in E for the left move is: X − ⊲ φ ( X, Y ) . X − ⊲ ψ ( X, Z ) = 1 E with Z = X − ∂φ ( X, Y ) − Y X , which is the 1st equality in (16). Applying ∂ , it follows that W = Y .The equation in E for the move on the right: U − ⊲ ψ ( Y, T ) . X − ⊲ φ ( Y, X ) = 1 E with T = Y − ∂φ ( Y, X ) − XY and U = ∂ψ ( Y, T ) − Y T Y − , follows from the 2nd equality of (16) for X = ∂ψ ( Y, T ) − Y T Y − , and therefore T = Y − ∂ψ ( Y, T ) XY , or T = Y − ∂φ ( Y, X ) − XY , by actingwith X − = U − .R2C: Fix X, Z ∈ G . The corresponding equation in E : Y − ⊲ φ ( X, Y ) . Y − ⊲ ψ ( X, Z ) = 1 E = Z − ⊲ ψ ( Z, X ) . Z − ⊲ φ ( Z, T ) (17)with Y = ∂ψ ( X, Z ) − XZX − and T = ∂ψ ( Z, X ) − ZXZ − , is equivalent to (16) with Y substitutedby Z in the 2nd equality. Applying ∂ to (17), it follows that W = Z and U = X .R3: Fix X, Y, T ∈ G . The corresponding equation in E is the Reidemeister 3 equation (13), whichalso implies the equality U = Z , by applying ∂ .Invariance under the identity and interchange moves of Figure 7 is immediate - to show the latterwe use the interchange law for the operations of Figures 2 and 3.For framed tangles we need to show invariance of I Φ under the R ′ move using the properties (i)and (ii) of Definition 3.15, which replace the Reidemeister 1 condition (11).Fix Z ∈ G . For the move on the left, at the lower crossing we have, from (i), the relation Y = g ( Z ),and at the upper crossing we have the relation X = f ( g ( Z )) = Z , by (ii). The equation for the move,which is ψ ( Z, Z ) φ ( Z, Y ) = 1 E , follows from the 2nd equality in (16). For the move on the right, at thelower crossing we have, from (i), the relation V = f ( Z ), and at the upper crossing we have the relation W = g ( f ( Z )) = Z , by (ii). The equation for the move, namely φ ( V, Z ) ψ ( V, V ) = 1 E follows from the1st equality in (16).We close this section by stating a TQFT property of the invariant I Φ , which follows easily from thedefinition. Theorem 3.20
Let D and D be tangle diagrams, so that the vertical composition D D is well defined.For any enhancements ω and ω ′′ of the top of D and the bottom of D we have: (cid:28) ω (cid:12)(cid:12)(cid:12)(cid:12) I Φ (cid:18) D D (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ω ′′ (cid:29) = X ω ′ h ω | I Φ ( D ) | ω ′ ih ω ′ | I Φ ( D ) | ω ′′ i , where the sum extends over all possible enhancements ω ′ of the intersection of D with D . Recall that a rack R is given by a set R together with two (by the axioms not independent) operations( x, y ) ∈ R × R x ⊲ y ∈ R and ( x, y ) ∈ R × R x ⊳ y ∈ R , such that for each x, y, z ∈ R :1. x ⊲ ( y ⊳ x ) = y,
2. ( x ⊲ y ) ⊳ x = y, x ⊲ ( y ⊲ z ) = ( x ⊲ y ) ⊲ ( x ⊲ z ) ,
4. ( x ⊳ y ) ⊳ z = ( x ⊳ z ) ⊳ ( y ⊳ z ).12 ⊲ x yy x y x ⊳ yx y Figure 12: A rack colouring of a link diagram in the vicinity of a vertex.A quandle Q is a rack satisfying the extra condition: x ⊳ x = x = x ⊲ x , ∀ x ∈ Q .There is a more economical definition of a rack: it is a set R with an operation ( x, y ) x ⊳ y , suchthat condition (4) above holds and such that for each y ∈ R the map x x ⊳ y is bijective. Its inversewill give us the map x y ⊲ x . The following result and proof appeared in [23]. Lemma 4.1 (Nelson Lemma)
Given a rack R , the maps x x ⊲ x and x x ⊳ x are injective (thusbijective if the rack is finite.) Proof.
Let x and y belong to R . Then( x ⊲ x ) ⊲ y = ( x ⊲ x ) ⊲ ( x ⊲ ( y ⊳ x )) = x ⊲ ( x ⊲ ( y ⊳ x )) = x ⊲ y. If x ⊲ x = y ⊲ y then x ⊲ x = ( x ⊲ x ) ⊲ x = ( y ⊲ y ) ⊲ x = y ⊲ x. Since y ⊲ y = x ⊲ x then y ⊲ y = y ⊲ x . Thisimplies x = y , since the map x y ⊲ x is bijective, its inverse being x x ⊳ y . The proof for x x ⊳ x is analogous.Given a knot diagram D , and a rack R , a rack colouring of D is an assignment of an element of R to each arc of D , which at each crossing of the projection has the form shown in figure 12.The following is well known. Theorem 4.2
Let R be a finite rack. Then the number I R ( D ) of rack colourings of a link diagram D is invariant under the Reidemeister moves 1’, 2 and 3, and is therefore an invariant of framed links,also called I R . Moreover if R is a quandle then the number of rack colourings is invariant under theReidemeister 1 move, therefore defining a link invariant. For a proof see [23] or [17]. (Invariance under the Reidemeister moves 2 and 3 is immediate. Invari-ance under the Reidemeister 1’ move, which is less trivial, is a consequence of the Nelson Lemma.)
Let us see that rack and quandle invariants can be written in the framework of this article. Let R be arack, which we suppose to be finite. Consider an arbitrary group structure on R . Call the group G . Wedo not impose any compatibility relation with the rack operations, we just assume that the underlyingset of R coincides with the underlying set of G . Consider the crossed module G = (id : G → G, ad),where ad denotes the adjoint action of G on G . Put: ψ ( b, a ) = b a b − ( b ⊲ a ) − , φ ( b, a ) = a b ( a ⊳ b ) − b − . (18) Theorem 4.3
The pair
Φ = ( ψ, φ ) is a framed Reidemeister pair. Moreover Φ is an unframed Reide-meister pair if R is a quandle. Proof.
In this case relation R φ ( b, a ) ψ ( b, a ⊳ b ) = 1 which follows tautologically. The relation R φ ( b, a ) . bφ ( c, a ⊳ b ) b − . φ ( c, b ) = aφ ( c, b ) a − . φ ( c, a ) . cφ ( b ⊳ c, a ⊳ c ) c − , and follows easily, from relation (3) of the definition of a rack.13et us now prove relations 1. and 2. of Definition 3.15. Let a ∈ G . The equation z = ∂ ( φ ( a, z )) a means z = z a ( z ⊳ a ) − , or z ⊳ a = a , that is z = a ⊲ a . By the Nelson Lemma 4.1, for each z the equation z = a ⊲ a has a unique solution f ( z ) ∈ R . In this case g ( a ) = ∂ ( ψ ( a, a )) − a = a ⊲ a . Thus trivially f ◦ g = g ◦ f = id R .Finally if R is a quandle then ψ ( x, x ) = x ( x ⊲ x ) − = x x − = 1 E .Since there is clearly, by (8), (9) and (10), a one-to-one correspondence between G -colourings of alink diagram D and rack colourings (with respect to R ) of D , we have: Theorem 4.4
Given a link diagram D , we have I Φ ( D ) = I R ( D )1 G , where G is the identity of G .Therefore the class of invariants defined in this paper is at least as strong as the class of rack linkinvariants. There is a spin-off of the rack invariant in order to handle tangles. Given a rack R , recall that an R -enhanced tangle, Definition 3.8, is a tangle together with a map from the boundary of T into (theunderlying set of) R . There is a category whose objects are the words ω in R ⊔ R ∗ and whose morphismsare R -enhanced tangles connecting them. Thus if the word ω is the source of T then ω is a word having i elements, where i is the number of intersections of the tangle with R ×{ } . Moreover the n th element of ω is either the colour a ∈ R given to the n -th intersection, or it is a ∗ , the former happening if the strandis pointing downwards and the latter if the associated strand is pointing upwards. These conventionswere explained in figure 8. Given an R -enhanced tangle T let ω ( T ) and ω ′ ( T ) be the source and targetof T , both words in R ⊔ R ∗ . Define h ω ( T ) | I R ( T ) | ω ′ ( T ) i as being the number of arc-colourings of adiagram of T extending the enhancement of T . This defines an invariant of tangles, for any choice ofcolourings on the top and bottom of T (in other words, for any R -enhancement of T ). Clearly Theorem 4.5
For any R -enhanced tangle T , putting ω = ω ( T ) and ω ′ = ω ′ ( T ) we have (see definitions3.9 and 3.11 for notation): h ω ( T ) | I Φ ( T ) | ω ′ ( T ) i = h ω | I R ( T ) | ω ′ i e ( ω ) y e ( ω ′ ) e ( ω ) − e ( ω ′ ) . We can extend the statement of Theorem 4.4 for the case of rack cohomology invariants of knots. Let R be a rack. Let V be an abelian group. We say that a map w : R × R → V is a rack 2-cocyle if: w ( x, y ) + w ( x ⊳ y, z ) = w ( x, z ) + w ( x ⊳ z, y ⊳ z ) , for each x, y, z ∈ R. If R is a quandle, such a w is said to be a quandle cocycle if moroever w ( x, x ) = 0 V , for each x ∈ R .For details see [10, 9, 23, 12, 13].Consider any group structure G on the set R , which may be completely independent of the rackoperations. Consider the crossed module ( ∂ : G × V → G, • ), where g • ( h, v ) = ( ghg − , v ) , for each g, h ∈ G and v ∈ V, which is a left action of G on G × V by automorphisms, and ∂ ( g, v ) = g , for each( g, v ) ∈ G × V . Given a rack 2-cocycle w : R × R → V , set: ψ ( b, a ) = (cid:0) b a b − ( b ⊲ a ) − , w ( b ⊲ a, b ) (cid:1) , φ ( b, a ) = (cid:0) a b ( a ⊳ b ) − b − , w ( a, b ) − (cid:1) . Theorem 4.6
The pair
Φ = ( ψ, φ ) is a framed Reidemeister pair. Its associated framed link invariantcoincides with the usual rack cohomology invariant of framed links. Morever if R is a quandle and w a quandle 2-cocycle then Φ = ( ψ, φ ) is an unframed Reidemeister pair and its associated invariant oflinks coincides with the usual quandle cocycle link invariants [10]. Proof.
Analogous to the proof of Theorems 4.3 and 4.4.Therefore the class of invariants defined in this paper is at least as strong as the class of invariants oflinks derived from quandle cohomology classes. 14 L K ′ L K K ′ Figure 13: Turning an oriented knot into a string knot in two different cases.
Recall that an (oriented) long knot is an embedding f of R into R such that, for sufficiently large (inabsolute value) t , we have f ( t ) = (0 , , − t ). These are considered up to isotopy with compact support.Clearly long knots (up to isotopy) are in one-to-one correspondence with isotopy classes of tangles whoseunderlying 1-manifold is the interval, and whose boundary is { } × { } × {± } , being, furthermore,oriented downwards. These are usually called string knots.There exists an obvious closing map, cl, sending a string knot L to a closed knot cl( L ). It is wellknown that this defines a one-to-one correspondence between isotopy classes of string knots and isotopyclasses of oriented knots. To see this, note that a map sending a closed knot K to a long knot L K canbe obtained by choosing a base point p ∈ K . Then there exists an (essentially unique) orientation (of S and of K ) preserving diffeomorphism ( S \ { p } , K \ { p } ) → ( R , L pK ), where L pK is a long knot withcl( L pK ) = K . Note that L pK depends only on the orientation preserving diffeomorphism class of thetriple ( S , K, p ), thus since all pairs ( K, p ), with fixed K , but arbitrary p , are isotopic we can see that L pK depends only on K , thus we can write it as L K .Let D be a knot diagram of the knot K . Consider the Wirtinger generators of the fundamentalgroup π ( C K ) of the complement C K = S \ n ( K ) of K (here n ( K ) is an open regular neighbourhood of K ); we thus have a meridian for any arc of the diagram D . Let a be an arc of D and p a base point of K in a . Then there is a meridian m p = m of K encircling the arc a at p , whose direction is determined bythe right hand rule. Let D = { z ∈ C : | z | ≤ } and S = ∂D . Choose an embedding f : S × D → S such that • f ( S × { } ) = K , preserving orientations, with f (1 ,
0) = p . • f ( { } × S ) = m • f ( S × { } ) has zero linking number with K .If we take f (1 ,
1) to be the base point of S , then the homotopy class l p = l ∈ π ( C K ) of f ( S × { } ) iscalled a longitude of K [8]. It is well known that the triple ( π ( C K ) , m, l ), considered up to isomorphism,is a complete invariant of the knot K [29]. Note that if we choose another base point p ′ of K then m p ′ and l p ′ can be obtained from m p and l p by conjugating by a single element of π ( C K ).The longitude l p , being an element of the fundamental group of the complement C K of K , cancertainly be expressed in terms of the Wirtinger generators. This can be done in the following way; fordetails see [12, 13]. Let a = a be the arc of the diagram D of K containing the base point of K . Thengo around the knot in the direction of its orientation. This makes it possible to order the arcs of K ,say as a , a , . . . , a n ; we would have a n = a , except that we prefer to see K as being split at the basepoint p , separating the arc a in two. We can also order the crossings of D .The longitude l p of K is expressed as a product of all elements of π ( C K ), associated to the arcs weundercross as we travel from p to p , making sure that the linking number of l p with K is zero. Thereforeany arc a i has also assigned a partial longitude l i (the product of the elements of π ( C K ), associated tothe arcs we undercross, as we travel from p to a i ). We thus have l n = l . Given an arc a of D denotethe corresponding element of the fundamental group of the complement by g a . Then clearly we havethat g a i = l − i g a l i . The way to pass from l i to l i +1 appears in figure 14 for the positive and negativecrossing. 15 j a i a i +1 a j a i a i +1 l i = g a l i +1 l − j g − a l l j = g a l i +1 g − a j l i +1 = l i g − a i g a j l i = g − a l i +1 l − j g a l l j = g − a l i +1 g a j l i +1 = l i g a i g − a j or orFigure 14: Rules for partial longitudes at crossings.Given i , let j i be the number of the arc splitting a i and a i +1 . Let θ i be the sign of the i -th crossing.Then: l = n − Y i =1 g − θ i a i g θ i a ji = n − Y i =1 l − i g − θ i a l i l − j i g θ i a l j i = n − Y i =1 [ l − i , g − θ i a ] [ g − θ i a , l − j i ]; (19)more generally, if k ∈ { , . . . , n } : l k = k − Y i =1 g − θ i a i g θ i a ji = k − Y i =1 l − i g − θ i a l i l − j i g θ i a l j i = k − Y i =1 [ l − i , g − θ i a ] [ g − θ i a , l − j i ] . (20)Thus both the longitude l and any partial longitude l k belong to the commutator subgroup of thefundamental group of the complement of K . Also l i +1 = l i [ l − i , g − θ i a ] [ g − θ i a , l − j i ] . In remark 4.13 we will present another formula for a knot longitude.
Let K be a knot in S . Consider the fundamental group of the complement C K = S \ n ( K ) of the knot K . Here n ( K ) is an open regular neighbourhood of K . Choose a base point p of K . Let the associatedmeridian and longitude of K in π ( C K ) be denoted by m p and l p , respectively. Note that [ m p , l p ] = 1.Let f : π ( C K ) → G be a group morphism. Therefore f ( l p ) ∈ G ′ . = [ G, G ], the derived (commutator)group of G , generated by the commutators [ g, h ] . = ghg − h − . Moreover [ f ( l p ) , f ( m p )] = 1 G . Then f ( l p ) ∈ Λ . = [ G, G ] ∩ C ( x ) , where x = f ( m p ) and C ( x ) is the set of elements of G commuting with x .Let G be a finite group. Let x be an element of G . The Eisermann invariant [13] (also calledEisermann polynomial) is: E ( K ) = X { f : π ( C K ) → G | f ( m p )= x } f ( l p ) ∈ N (Λ) . Note that if we choose a different base point p ′ of K then E ( K ) stays invariant since m p ′ = h m p h − and l p ′ = h l p h − , for some common h ∈ π ( C K ). The Eisermann invariant can be used to detect chiraland non invertible knots [13].Clearly E ( K ) is given by a map f KE : G ′ → N , where E ( K ) = X g ∈ G ′ f KE ( g ) g. Note that f KE ( g ) = 0 if g Λ. 16et us see that the Eisermann invariant can be addressed using Reidemeister pairs. This is aconsequence of the previous subsections and the discussion in [13, 12], which we closely follow, havingdiscussed and completed the most relevant issues in 4.2.1.Let G be a group. Choose x ∈ G and consider from now on the pair ( G, x ). We will use the notation h g = g − hg , where g, h ∈ G . Let Q = (cid:8) x g , g ∈ G ′ (cid:9) ⊂ G, Q = (cid:8) x g , g ∈ G (cid:9) ⊂ G. Lemma 4.7
Both sets Q and Q are self conjugation invariant: a, b ∈ Q = ⇒ a − ba ∈ Q and a, b ∈ Q = ⇒ a − ba ∈ Q Therefore Q and Q are both quandles, with quandle operation h ⊳ g = h g . Proof.
Given g, h ∈ G ′ we have( g − xg ) − ( h − xh )( g − xg ) = x x − hg − xg , and x − hg − xg = x − hg − xgh − h = [ x − , hg − ] h ∈ G ′ . The proof for Q is analogous.It is easy to see that: Lemma 4.8 (Eisermann)
Let G be a group. Given arbitrary x ∈ G , both G ′ and G are quandles,with quandle operation: h ⊳ g = x − hg − xg , g ⊲ h ′ = xh ′ g − x − g. (21) There are also quandle maps p : G ′ → Q and p : Q → G with p ( g ) = x g . Recall the rack invariant of tangles, defined just before Theorem 4.5.
Theorem 4.9 (Eisermann)
For any knot K and any g in G ′ it holds that: h G | I G ′ ( D K ) | g i = f KE ( g ) , where D K is any string knot diagram associated to K . Of course we regard D K as a G -enhanced tanglediagram coloured with G = 1 G ′ at the top and with g at the bottom. Proof.
Follows from the discussion in 4.2.1 and especially figure 14.
Remark 4.10
The previous theorem is also valid for the quandle structure in G , Lemma 4.8. Giventhe form of the quandle is easy to see that if g, h ∈ G ′ : h g | I G ′ ( D K ) | h i = h g | I G ( D K ) | h i . By using subsection 4.1 we will show that the Eisermann invariant can be addressed in our framework,by passing to string knots. Suppose we are given a finite group G and x ∈ G (it may be that x ∈ G \ G ′ ).We can choose any group operation structure in the underlying set of G ′ . We take the most obviousone, given by the inclusion G ′ ⊂ G . The associated crossed module is G ′ id → G ′ , with G ′ acting on itselfby conjugation.Define, given g, h ∈ G ′ , the pair Φ x = ( ψ x , φ x ), as: φ x ( g, h ) = hg ( x g ) − x h h − g − = hx − gh − xg − = [ hx − , gx − ] ψ x ( g, h ) = [ g, h ][ hg − , x ] = [ xhg − x − gx − , gx − ] − (22)17 heorem 4.11 The pair Φ x = ( ψ x , φ x ) is an unframed Reidemeister pair for the crossed module G ′ id → G ′ , with G ′ acting on itself by conjugation. Let K be an oriented knot and L K be the associated stringknot. Given g ∈ G ′ then h G ′ | I Φ x ( L K ) | g i = f KE ( g ) . (23) Proof.
The expressions for Φ x = ( ψ x , φ x ) guarantee that the colourings of the arcs at a crossing arethose given by the Eisermann quandle operation and its inverse (Lemma 4.8). Thus equation (23) holds,and it is enough to check that Φ does indeed satisfy the conditions to be a unframed Reidemeister pair.Clearly the Reidemeister 1 condition (11) holds: ψ x ( l, l ) = 1. The Reidemeister 2 equation (12) is: φ x ( l, m ) ψ x ( l, x − ml − xl ) = 1, i.e.[ mx − , lx − ] [ l, x − ml − xl ] [ x − ml − x, x ] = 1 . Writing this out in full, one obtains: mx − lx − xm − xl − . l ( x − ml − xl ) l − ( l − x − lm − x ) . ( x − ml − x ) x ( x − lm − x ) x − = m x − lm − xl − . lx − ml − x l − x − lm − x . x − ml − xl m − = 1 . In the above computation, the underlined factors are equal to 1.To avoid confusion with the quandle operation ⊲ , the left action of G ′ on G ′ by conjugation will bedenoted by g • h , thus g • h = ghg − for each g, h ∈ G ′ . The Reidemeister 3 equation (13) for this casereads: φ x ( l, m ) . l • φ x ( n, p ) . φ x ( n, l ) = m • φ x ( n, l ) . φ x ( n, m ) . n • φ x ( r, q ) , where p = x − ml − xl, r = x − ln − xn, q = x − mn − xn. The left-hand side of the above equation,written out in full, is: mx − lm − xl − . l ( px − np − xn − ) l − . ( lx − nl − xn − )= m x − lm − x . x − ml − x lx − n ( l − x − lm − x ) xn − . x − nl − xn − = mln ( n − x − nl − x − lm − x )( n − x − nl − x ) n − , and the right-hand side leads to the same expression: m ( lx − nl − xn − ) m − . mx − nm − xn − . n ( qx − rq − xr − ) n − = mlx − nl − xn − x − nm − x ( x − mn − xn ) . x − ( x − l n − xn )( n − x − n m − x ) x ( n − x − nl − x ) n − = mln ( n − x − nl − x − lm − x )( n − x − nl − x ) n − . In both derivations we insert the definitions of p, r and q in the 1st equality, and regroup the factors aftereliminating the underlined expressions in the 2nd equality. Note that both sides equal mlnu − r − n − ,which is the product of the colourings of the 6 external arcs in Figure 15 for Reidemeister 3, taken inanticlockwise order starting with m , where u is the colouring assigned to the rightmost upper arc, i.e. u = x − pn − xn = x − qr − xr .Consider the quandle structure in G , Lemma 4.8, given by the same formulae as the one of G ′ .Consider the crossed module given by the identity map G → G and the adjoint action of G on G . Given x ∈ G we have an unframed Reidemeister pair ¯Φ x = ( ¯ ψ x , ¯ φ x ), with the same formulae as (22), namely:¯ φ x ( g, h ) = hg ( x g ) − x h h − g − = hx − gh − xg − = [ hx − , gx − ] , ¯ ψ x ( g, h ) = [ g, h ][ hg − , x ] = [ xhg − x − gx − , gx − ] − (24)for g, h ∈ G . It is easy to see that: Proposition 4.12
Let a, b ∈ G ′ . For each x ∈ G , and each string knot L K : h a | I Φ x ( L K ) | b i = h a | I ¯Φ x ( L K ) | b i l nn r Up m l nn r uq Figure 15: Two sides of the Reidemeister-III move in the proof of Theorem 4.11. hg xhg − x − gx gh − x − hg − x − gx hg − x − gh − x − hg − x − g h g x − hg − xgx − gh − xhg − xgx − hg − xgh − xhg − xg Figure 16: The Eisermann polynomial for the positive and negative trefoil knots K + and K − Remark 4.13 (Formula for a knot longitude)
Our approach for defining Eisermann invariants,and the proof of Theorem 4.11, provides a different formula to (19) for the longitude l p = l of a knot K ,if K is presented as the closure of a string knot L . It is assumed that the base point p ∈ K of the closedknot K lives in the top end of L . Let G = π ( C K ) . Consider the crossed module (id : G → G, ad) . Let x be the element of G given by the top strand a of L . Let b be the bottom strand of L . Consider theReidemeister pair ¯Φ x in (24) , thus ¯ φ x ( g, h ) = [ hx − , gx − ] and ¯ ψ x ( g, h ) = [ g, h ][ hg − , x ] = [ xhg − x − gx − , gx − ] − . (25) Consider a diagram D of L . Colour each arc c of the diagram D with the corresponding partial longitude l c , as defined in 4.2.1. Therefore l a = 1 and l b = l . Then by the proof of Theorem 4.11 one has aReidemeister colouring F . If we evaluate F (definitions 3.9 and 3.11), we have a morphism l a e ( F ) −−−→ l b ,hence e ( F ) = l = l p . The form of e ( F ) thus yields an alternative formula for the knot longitude, whichwill be crucial for giving a homotopy interpretation of the lifting (Theorem 4.17) of the Eisermanninvariant. Let G be a group with a base point x . The explicit calculation of the invariant h a | I ¯Φ x ( K + ) | b i and h a | I ¯Φ x ( K − ) | b i for the trefoil knot K + and its mirror image K − (the positive and negative trefoils),converted to string knots, appears in figure 16. In particular, given a, g, h ∈ G , we have: h a | I ¯Φ x ( K + ) | g i = (cid:8) h ∈ G : x hg − x − gh − x − hg − x − g = a ; x gh − x − hg − x − g = h (cid:9) , h a | I ¯Φ x ( K − ) | h i = (cid:8) g ∈ G : x − hg − xgh − xhg − xg = g ; x − gh − xhg − xg = a (cid:9) . Consider from now on G = S . We refer to table 1, displaying the values of h | I ¯Φ x ( K − ) | a i and h | I ¯Φ x ( K + ) | a i for some choices of x ∈ S , representing all possible conjugacy classes in S . In each 2nd19 id (12) (12)(34) (123) (123)(45) (1234) (12345) K − id 7id 5id 7id id id + 4(13)(24) id + 5(12345) K + id 7id 5id 7id id id + 4(13)(24) id + 5(15432)Table 1: The Eisermann invariant for the negative and positive trefoils for G = S .and 3rd row entry of table 1, we put: X h ∈ S h | I ¯Φ x ( K − ) | h i and X g ∈ S h | I ¯Φ x ( K + ) | g i , both elements of the group algebra of S . Therefore the invariant L
7→ h | I ¯Φ x ( L ) | b i , where we identifya knot with its associated string knot, separates the trefoils, for x = (12345). This is due to Eisermann[13]. Recall the construction of the Eisermann invariant in our framework (subsection 4.2). The quandleunderlying the Eisermann invariant corresponds to the Reidemeister pair given in equation (22).
Definition 4.14 (Unframed Eisermann lifting )
Let G be a finite group and x ∈ G . An unframedEisermann lifting is given by a crossed module ( ∂ : E → G, ⊲ ) , and an unframed Reidemeister pair Φ x = ( φ x , ψ x ) , where φ x , ψ x : G × G → E , such that, given L, M ∈ G : ∂ ( φ x ( L, M )) = [
M x − , Lx − ] , ∂ ( ψ x ( L, M )) =[
L, M ][ M L − , x ] . The colourings of the arcs of any tangle diagram will correspond to those given by the Eisermannquandle, but there may be additional information contained in the assignments of elements of E to thecrossings of the diagram, i.e. an Eisermann lifting is a refinement of the Eisermann invariant.In this subsection we will construct an unframed Eisermann lifting from each central extension ofgroups: { } → A → E ∂ −→ G → { } . Here G is a finite group and ∂ : E → G is a surjective group map, such that the kernel A of ∂ is centralin E .Choose an arbitrary section s : G → E of ∂ , meaning ∂ ( s ( g )) = g , for each g ∈ G . Therefore s ( gh ) = s ( g ) s ( h ) λ ( g, h ), where λ ( g, h ) is in the centre of E , for each g, h ∈ G . Moreover given e ∈ E then s ( ∂ ( e )) = e c ( e ), where c ( e ) is in the centre of E . This is because ∂ ( s ( ∂ ( e )) = ∂ ( e ). Clearly: Lemma 4.15
The map ( g, e ) ∈ G × E g ⊲ e = s ( g ) e s ( g ) − ∈ E is a left action of G on E byautomorphisms, and with this action ∂ : E → G is a crossed module. Moreover, the action ⊲ does notdepend on the section s . Given a section s : G → E of ∂ : E → G , define, for each g, h ∈ G : { g, h } = [ s ( g ) , s ( h )] . (26)(This does not depend on the chosen section s of ∂ since ker( ∂ ) is central in E .) Lemma 4.16
For each a, b ∈ E we have [ a, b ] = { ∂ ( a ) , ∂ ( b ) } . Proof.
Given a, b ∈ E we have { ∂ ( a ) , ∂ ( b ) } = [ s ( ∂ ( a )) , s ( ∂ ( b )] = [ a c ( a ) , b c ( b )] = [ a, b ] . heorem 4.17 Let G be a finite group and x ∈ G . Let ∂ : E → G be a surjective group morphism suchthat the kernel A of ∂ is central in E . The pair Φ x = ( φ x , ψ x ) , given by: φ x ( g, h ) = { hx − , gx − } ψ x ( g, h ) = { g, h }{ hg − , x } ; is an unframed Eisermann lifting for the crossed module ( ∂ : E → G, ⊲ ) , of Lemma 4.15. Proof.
Given
L, M ∈ G , then ∂ ( { L, M } ) = ∂ ([ s ( L ) , s ( M )]) = [ ∂ ( s ( L )) , ∂ ( s ( M ))] = [ L, M ]. Since ∂ issurjective we can find l ∈ E such that L = ∂ ( l ), and likewise M = ∂ ( m ) and x = ∂ ( y ). The Reidemeister2 condition (12), which is: { M x − , Lx − } { L, x − M L − xL } { x − M L − x, x } = 1 , becomes { ∂ ( my − ) , ∂ ( ly − ) } { ∂ ( l ) , ∂ ( y − ml − yl ) } { ∂ ( y − ml − y ) , ∂ ( y ) } = 1 , or, what is the same (by using lemma 4.16):[ my − , ly − ] [ l, y − ml − yl ] [ y − ml − y, y ] = 1 . This is an algebraic identity which was shown to hold in the proof of Theorem 4.11. An analogousargument shows that the Reidemeister 3 equation (13) is satisfied, since ∂ ( l ) ⊲ m = lml − , so that wecan use the algebraic identity for Reidemeister 3 from the same proof. For the Reidemeister 1 move thisfollows from ψ x ( M, M ) = { M x − , M x − } = { ∂ ( my − ) , ∂ ( my − ) } = [ my − , my − ] = 1 E . Remark 4.18
By the proof of the previous theorem, we can see that an alternative expression for ψ x is: ψ x ( L, M ) = { xM L − x − Lx − , Lx − } − . Note that for each
L, M, x ∈ E we have (since these are identities between usual commutators) that { xM L − x − Lx − , Lx − } − = { L, M }{ M L − , x } . In the context of Theorem 4.17, let us find a lifting of the Eisermann invariant for the case of G = S ,for which we gave detailed calculations in subsection 4.2. It is well known that S is isomorphic toPGL(2 , Z , modulo the central subgroup Z ∗ of diagonal matrices which are multiples of the identity. We thus have a central extension: { } → Z ∗ i −→ GL(2 , p −→ PGL(2 , ∼ = S → { } . Here GL(2 ,
5) is the group of invertible two-by-two matrices in the field Z .Let K + and K − be the right and left handed trefoils. In table 2 we display h a | I Φ x ( K + ) | i and h a | I Φ x ( K − ) | i for some choices of x ∈ PGL(2 , ∼ = S , representing all possible conjugacy classes in S ∼ = PGL(2 , X s ∈ PGL(2 , h s | I Φ x ( K + ) | i and X s ∈ PGL(2 , h s | I Φ x ( K − ) | i , respectively, both elements of the group algebra of GL(2 , A ∈ GL(2 , , e A .Comparing with table 3, which shows the unlifted Eisermann invariant I Φ x for G = PGL(2 , ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) K + (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) + 6 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) + 4 (cid:18) (cid:19) (cid:18) (cid:19) + 5 (cid:18) (cid:19) K − (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) + 6 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) + 4 (cid:18) (cid:19) (cid:18) (cid:19) + 5 (cid:18) (cid:19) Table 2: The lifted Eisermann invariant for the positive and negative trefoils in 4.3.1 x ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) K + ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) + 4 ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) + 5 ^ (cid:18) (cid:19) K − ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) + 4 ^ (cid:18) (cid:19) ^ (cid:18) (cid:19) + 5 ^ (cid:18) (cid:19) Table 3: The unlifted Eisermann invariant for the positive and negative trefoils in 4.3.1itself. Specifically, looking at the penultimate column of tables 2 and 3, thus x = ^ (cid:18) (cid:19) , we cansee that the lifting distinguishes the trefoil from its mirror image. Namely, for the lifted Eisermanninvariant, noting: ^ (cid:18) (cid:19) = ^ (cid:18) (cid:19) = ^ (cid:18) (cid:19) , we have: * ^ (cid:18) (cid:19)(cid:12)(cid:12)(cid:12) I Φ x ( K + ) (cid:12)(cid:12)(cid:12) ^ (cid:18) (cid:19)+ = 4 (cid:18) (cid:19) = 4 (cid:18) (cid:19) = * ^ (cid:18) (cid:19)(cid:12)(cid:12)(cid:12) I Φ x ( K − ) (cid:12)(cid:12)(cid:12) ^ (cid:18) (cid:19)+ , whereas for the unlifted Eisermann invariant I Φ x : * ^ (cid:18) (cid:19)(cid:12)(cid:12)(cid:12) I Φ x ( K + ) (cid:12)(cid:12)(cid:12) ^ (cid:18) (cid:19)+ = 4 ^ (cid:18) (cid:19) = * ^ (cid:18) (cid:19)(cid:12)(cid:12)(cid:12) I Φ x ( K − ) (cid:12)(cid:12)(cid:12) ^ (cid:18) (cid:19)+ . Table 3 should be compared with table 1.
To give a homotopy interpretation of the lifting of the Eisermann invariant, Theorem 4.17, we recallthe notion of non-abelian tensor product and wedge product of groups, due to Brown and Loday [6, 7];see also [5]. Let G be a group. We define the group G ⊗ G (a special case of the tensor product oftwo groups G ⊗ H ) as being the group generated by the symbols g ⊗ h , where g, h ∈ G , subject to therelations, ∀ g, h, k ∈ G : gh ⊗ k = ( ghg − ⊗ gkg − ) ( g ⊗ k ) , (27) g ⊗ hk = ( g ⊗ h ) ( hgh − ⊗ hkh − ) . (28)The key fact about the non-abelian tensor product of groups is that there is a homomorphism ofgroups δ : G ⊗ G → G ′ = [ G, G ], defined on generators by g ⊗ h [ g, h ], which is clearly surjective.Surjectivity also holds if we replace G ⊗ G by the group G ∧ G , obtained from G ⊗ G by imposing theadditional relations: g ⊗ g = 1 , ∀ g ∈ G.
22e denote the image of g ⊗ h in G ∧ G by g ∧ h . Finally [5], there is a left action • by automorphismsof G on G ⊗ G and G ∧ G , given by: g • ( h ⊗ k ) = ( ghg − ) ⊗ ( gkg − ) and g • ( h ∧ k ) = ( ghg − ) ∧ ( gkg − ) , and we have two crossed modules of groups: ( δ : G ⊗ G → G ′ , • ) and ( δ : G ∧ G → G ′ , • ). This fact(which is not immediate) is Proposition 2.5 of [7].The following theorem is fully proved in [7] and [22]. Group homology is taken with coefficients in Z . Theorem 4.19 (Brown-Loday / Miller)
Let G be a group. One has an exact sequence: { } → H ( G ) → G ∧ G δ −→ G ′ → { } . Consider a central extension of groups { } → A → E ∂ −→ G → { } , where G is finite. Let K be aknot. Let C K be its complement. Then it is well known, and follows from the asphericity of the knotcomplement C K [25] (which is therefore an Eilenberg-MacLane space), combined with the fact that knotcomplements are homology circles [8], that H ( π ( C K )) = { } . Therefore, by the Brown-Loday / MillerTheorem we have: π ( C K ) ∧ π ( C K ) ∼ = [ π ( C K ) , π ( C K )] = π ( C K ) ′ , canonically. Let now f : π ( C K ) → G be a group morphism. Defineˆ f = π ( C K ) ∧ π ( C K ) → E, as acting on the generators x ∧ y of π ( C K ) ∧ π ( C K ) ∼ = [ π ( C K ) , π ( C K )] by ˆ f ( x ∧ y ) = { f ( y ) , f ( x ) } − ;see Lemma 4.16. That the map ˆ f respects the defining relations for the non-abelian wedge product,follows from the fact that ker( ∂ ) is central in E , as in the proof of Theorem 4.17.Going back to the knot K , choose a base point p ∈ K . Let m p ∈ π ( C K ) and l p ∈ π ( C K ) be theassociated meridian and longitude. Then l p ∈ [ π ( C K ) , π ( C K )] ∼ = π ( C K ) ∧ π ( C K ) . Given an element x ∈ G , we thus have a knot invariant of the form: X f : π ( C K ) → G with f ( m p )= x ˆ f ( l p ) ∈ Z [ E ] . Theorem 4.20
Given x ∈ G , a finite group, let Φ x be the unframed Reidemeister pair derived fromthe central extension of groups { } → A → E ∂ −→ G → { } ; Theorem 4.17. Let K be a knot, with a basepoint p . Let L K be the associated string knot. Then: X a ∈ G h G | I Φ x ( L K ) | a i = X f : π ( C K ) → G with f ( m p )= x ˆ f ( l p ) . Proof.
The proof is exactly the same as for the unlifted case. Note Remark 4.13.
Acknowledgements
J. Faria Martins was supported by CMA/FCT/UNL, under the grant PEst-OE/ MAT/UI0297/2011.This work was partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia through the projectsPTDC/MAT/098770/2008, PTDC/MAT/ 101503/2008, and PEst-OE/EEI/LA0009/2013. We wouldlike to thank Ronnie Brown for comments. 23 eferences [1] J.C. Baez, A.D. Lauda, Higher-dimensional algebra. V. 2-groups,
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