Polyak type equations for virtual arrow diagram formulas in the annulus
aa r X i v : . [ m a t h . G T ] M a y Polyak type equations for virtual arrow diagraminvariants in the annulus
Arnaud Mortier [email protected]
November 21, 2018
Abstract
We describe the space of arrow diagram formulas (defined in [13]) for virtualknot diagrams in the annulus R × S as the kernel of a linear map, inspiredfrom a conjecture due to M. Polyak. As a main application, we slightly improveGrishanov-Vassiliev’s theorem for planar chain invariants ([6]). Contents vs virtual invariants . . . . . . . . . . . . . . . . 52.3.2 Homogenous virtual invariants . . . . . . . . . . . . . . . 62.4 The Polyak algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Based and degenerate diagrams . . . . . . . . . . . . . . . . . . . 62.6 Polyak’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Gauss diagrams were introduced in knot theory for the purpose of extractingnew combinatorial data from the widely studied knot diagrams. On one hand itgave rise to a generalization of knot theory, known as virtual knot theory [7]. On1nother hand, it allowed a new point of view on Vassiliev’s finite type invariants(see [13], [4], [2]). Several approaches have been used in order to define finitetype invariants for virtual knots. Vassiliev-Kauffman’s invariants [7] are directlyinspired from the axiomatic definition of Vassiliev invariants given by J.Birmanand X.-S.Lin [1], while the approach of M.Goussarov, M.Polyak and O.Viro(GPV, [5]) is inspired from the representation of Vassiliev invariants due toGoussarov [4].Another direction of investigation is the approach of T.Fiedler, who deco-rates Gauss diagrams with homological information when the knot diagramslive in a surface that is more complicated than the sphere or the plane.Here we focus on homogeneous GPV’s invariants for virtual knot diagramsin the annulus. • The annulus, because it has an abelian fundamental group. This propertyallows one to prove that Fiedler’s decorated Gauss diagrams encode theknot diagrams faithfully – i.e. with no loss of information [8]. • Homogeneous GPV invariants, because as we will show it is the goodframework to consider a conjecture of M.Polyak, who predicts the exis-tence of a linear map whose kernel consists of Gauss diagram invariants.Every result in this paper can be actually extended to the case of an arbitrarysurface replacing the annulus (except for Theorem 3.6 where the surface needsto be orientable) but it requires more complicated combinatorial tools. It willbe done in a forthcoming paper.
Acknowledgements
The author thanks Thomas Fiedler for introducing him to the subject of Gaussdiagram invariants, and for useful remarks on the presentation. He also ac-knowledges useful corrections from Victoria Lebed, and thanks the referee forcareful reading and lucid remarks.
Warning . Though every Gauss or arrow diagram in this article comes withhomological markings due to the solid torus framework, we will often refer toworks where this is not the case, since many notions do not depend on this.Though it is not always explicitly mentioned, everything depends on the valueof a fixed integer K which is the global marking of every diagram (see section 2.1below). Following T. Fiedler ([2], [3]) we define a (decorated) Gauss diagram (of degree n) as an oriented circle marked with an integer, and n oriented chords (the arrows ,2hich are abstract, i.e. only the endpoints matter), each one equipped with asign (also writhe ) and an integer (its marking ), up to oriented homeomorphismsof the circle. It is to be understood that the 2 n endpoints of the arrows aredistinct. It is proved in [8] that such Gauss diagrams are in 1-1 correspondencewith virtual knot diagrams in the annulus, up to usual and virtual Reidemeistermoves. We denote by G n (resp. G ≤ n ) the Q -vector space freely generated byGauss diagrams of degree n (resp. ≤ n ), and set G = lim −→ G ≤ n .To the well-known Reidemeister moves for knot diagrams correspond R-moves for Gauss diagrams (see Fig.1 – as usual, the unseen parts must be thesame for all of the diagrams that belong to a given equation.). Beware thatthese moves depend on the homology class K of the considered knot diagrams.
1R R 3 a+bb + = a+ − a+bb+ + a− = a+b−Ka b a a+b−Kb+ + + +− − ε − ε − = ε ε a a a a =R 2= = +/− +/− K Figure 1: R-moves for decorated Gauss diagrams with global marking K We prove the following:
Theorem 2.1.
The equivalence class of a Gauss diagram associated with a knotdiagram, modulo the R -relations of Fig.1, is a complete invariant for virtualknots with homology class K . There is a linear isomorphism I : G ≤ n → G ≤ n that associates to a Gaussdiagram the formal sum of its subdiagrams (see [5]). A Gauss diagram formula is a knot invariant of the form G w ( G, I ( G )) , (1)where G ∈ G , G is the Gauss diagram associated with a knot projection and w is an orthogonal scalar product with respect to the basis of G given by Gaussdiagrams. Since this theory was born, mainly two scalar products have beenused, namely: • The orthonormal scalar product, which we shall denote by ( , ). It is usednotably in [5] and [2]. 3 Its normalized version h , i , defined by h G, G ′ i := | Aut( G ) | · ( G, G ′ ) , (2)where Aut( G ) is the set of symmetries of G , i.e. rotations that keep itunchanged (it is a subgroup of Z / n ).Roughly speaking, h , i counts parametrized configurations of arrows, while ( , )counts unordered sets of arrows. Notice that h , i is still symmetric (hence ascalar product). Obviously, the two definitions coincide when one deals withlong knots (and thus based Gauss diagrams).The second version was already defined in [13] (though their Theorem 2 isstated in terms of ( , )), but it was O.P. ¨Ostlund who first formally stated that h , i is more convenient to get nice properties when dealing with Gauss diagramswith symmetries ([9], sections 2 . . h , i is also used in [6], and implicitly in [15]. Take a Gauss diagram G and forget the signs associated with the arrows. Wecall what remains an arrow diagram (see [11]; beware that the terminology in[5] is different: arrows in arrow diagrams are signed). Arrow diagram spaces A n , A ≤ n and A , and the pairings ( , ), h , i are defined similarly to the previoussection.The raison d’ˆetre of this notion lies in the following map: take an ar-row diagram A ∈ A n and number its arrows from 1 to n . Then any map σ : { , . . . , n } → {± } defines a Gauss diagram A σ . Let sign( σ ) be the productof all the σ ( i )’s. We put: S ( A ) def = X σ ∈{± } n sign( σ ) · A σ . (3) S extends linearly into a map A n → G n . A Gauss diagram formula that lies inthe image of this map is called an arrow diagram formula . A lot of the explicitformulas that have been found so far are actually arrow diagram formulas – aswell in the framework of knots in S .Considering this map is relevant only in the context of the h , i pairing (2).Indeed, one may define (as most authors do) brackets (( A, G )) and hh A, G ii , with A ∈ A and G ∈ G in the following way: for every subdiagram ( i.e. unorderedset of arrows) of G that becomes A after one forgets its signs, form the productof these signs. Sum up all these products, and call the result (( A, G )). On theother hand, put hh A, G ii := | Aut( A ) | · (( A, G )). Then, of the naturally expectedrelations ((
A, G )) ? = ( S ( A ) , I ( G )) and hh A, G ii ? = h S ( A ) , I ( G ) i , A has no symmetries (see Lemma 4.1).A special interest arises in arrow diagram formulas in the case of virtual knottheory, as we shall see in the next subsection. Virtual knot theory arises as the natural “completion” of classical knot theorywith respect to Gauss diagrams. Indeed, while a knot diagram may be repre-sented by a Gauss diagram (with corresponding Reidemeister moves on Gaussdiagrams), a virtual knot diagram actually is a Gauss diagram. New (“virtual”)crossings are used as an artefact to draw planar representations of them, andthe additional virtual Reidemeister moves are precisely those planar moves thatdo not affect the underlying Gauss diagram (see [7]). vs virtual invariants One should be extremely cautious about the fact that the so-called “real” (orclassical) Reidemeister moves for Gauss diagrams may not always be actuallyperformed: for instance, two arrows may be added by Reidemeister II in the real settings only if the corresponding arcs of the knot diagram face each other– which seems not easy to check on the Gauss diagram.As a consequence, the framework introduced previously seems mostly com-fortable to look for virtual knot invariants.A natural related question is whether a given Gauss diagram formula for clas-sical knots always defines an invariant for virtual knots by the same equation (1).The answer is negative, the simplest example is the formula for the invariant v given by [13] (Theorem 2), which we reproduce with an example of noninvariance on Fig.2. v v v , < > = 0 + + = , < > + ++ = 1/2 = Figure 2: Polyak-Viro’s formula for v is not a virtual invariant5 .3.2 Homogenous virtual invariantsDefinition 2.2. For each n ∈ N , there is an orthogonal projection π n : G → G n with respect to the scalar product ( , ). For G ∈ G , there is some integer n suchthat G ∈ G ≤ n \ G ≤ n − . The principal part of G is defined by π n ( G ). G is called homogeneous if it is equal to its principal part. Lemma 2.3.
Let G ∈ G be a Gauss diagram formula for virtual knots. Thenits principal part lies in the image of the map S defined by (3), i.e. can berepresented by a (homogeneous) arrow polynomial. Corollary 2.4.
Any homogeneous Gauss diagram formula for virtual knots isan arrow diagram formula.
The above result in the context of knot theory in the sphere is contained inthe lines [5, section 3 . Theorem 2.5.
Let IA ≤ n be the space of arrow diagram formulas for virtualknots of degree no greater than n . Then: IA ≤ n = M k ≤ n ( IA ≤ n ∩ A k ) . A Gauss sum G ∈ G defines a virtual knot invariant if and only if the function h G, I ( . ) i is well defined on the quotient of G by Reidemeister moves on Gaussdiagrams. Hence it is interesting to understand the image of that subspaceunder the map I with a simple family of generators. This is the idea that ledthe construction of the Polyak algebra ([11, 5]) in the classical case. We adaptthis construction and define P as the quotient of G by the set of relations shownin Fig.3, which we call P , P and P (also 8 T ) relations for Gauss diagrams.The following theorem repeats Theorem 2 .D from [5] – the proof is similar. Theorem 2.6.
The map I induces an isomorphism G \ R → G \ P def = P , where R stands for the Reidemeister relations on Gauss diagrams. More precisely, I induces an isomorphism between Span(R i ) and Span(P i ) , for i = 1 , , . A based Gauss diagram is a Gauss diagram together with a distinguished ( base )arc on the circle, i.e. a region between two consecutive ends of arrows. Basedarrow diagrams are defined similarly. The corresponding spaces are denotedby G • and A • , in reference to the dot that we use in practice to pinpoint thedistinguished arc.A degenerate Gauss diagram (with one degeneracy) is a classical Gauss dia-gram in which one of the 2 n arcs of the base circle has been shrunk to a point.6 − ε − +++ ++ a+b−Ka+b−Ka+b−K a+b−Kb bb a a+b−K+ −+ + a+b−Kb a−+ + b−+ + a + + + +ba a −+a− −+ −b+ a+b+ − a+bb−a a+b+ a− a+b+ + a+b + + a+ − a+bb + + ab+ +−a a+bb+ + + + + or"8T" +/− K +/− ε a a
2P + + ε a a Figure 3: Relations defining the Polyak algebra in the solid torus frameworkWhen the arc was bounded by the two endpoints of one and the same arrow, thedegenerate diagram is decorated with the datum of which endpoint was beforethe other. In this way, there is a natural 1-1 correspondence between basedand degenerate diagrams. The spaces of degenerate diagrams are called DG and DA respectively. The latter is meant to be quotiented by the so-called trianglerelations , shown in Fig.4. The quotient space is denoted by DA / ∇ .This notion in the context of S is due to M. Polyak, and is part of hisconjecture which we discuss in the next section.7 += + b K−aa a+ba+bba a+b b K−ba a+b Figure 4: The triangle relations
During Swiss Knots conference in 2011, Michael Polyak gave a talk in whichhe conjectured that Gauss diagram formulas for knots in S were the space ofsolutions to an equation of type d ( · ) = 0 with d valued in DA / ∇ . We shall notgive a formal statement of this here, since it was never written by its author,but there is a video of the talk available online ([10]). The map d from thatconjecture has the property of being homogeneous: ∀ G ∈ G , [ d ( G ) = 0 ⇐⇒ ∀ n ∈ N , d ◦ π n ( G ) = 0] . It follows from 2.3.2 that, in the virtual setting, the best that we may expectfrom such a map is to detect arrow diagram formulas. For this reason, from nowon we mostly restrict our attention to this kind of invariants. We construct amap d in that framework, that differs from M.Polyak’s one by the signs in front ofthe contributing diagrams. Then we prove that its kernel encodes ReidemeisterIII invariance, while Reidemeister I and II are already very easy to check. Remark . Based on our understanding of the conjecture, Polyak-Viro’s for-mula for v (Fig.2) is a counterexample in the “classical” settings: it defines aninvariant of usual knots, but it does not have a trivial boundary, no matter howthe signs are chosen to compute it. Hence the present result seems to be thebest one can hope for. In this section we define a map d that will fit in some version of Polyak’s con-jecture for virtual knots in the solid torus.8 omogenous Polyak relations Let G ∈ G n . Then G satisfies the P i relations ( i.e. h G, P i i = 0)) if and onlyif it satisfies the homogeneous relations h G, π k (P i ) i = 0) for all k . These aredenoted by P , P ( n − , , P ( n − , , P ( n − , (or G T ) and P ( n − , (or G T ) –some examples are shown on Fig.5. The parenthesized numbers indicate in eachcase how many arrows are unseen. = ε − ε a a (n−2),2
2P 2P (n−1),1 ε − = + ε a a −b+ a+b+ − a+bb−a a+b+ a− a+b+ + a+b + + ab+ + + + −a a+bb+ + a+ − a+bb + = P 3 (n−3),3(n−2),2
P 3 = Figure 5: Some homogeneous Polyak relations
Lemma 3.1.
Let G ∈ G . Then G lies in the image of the map S if and only if G satisfies all the homogeneous relations D G, P ( n − , E = 0 . Homogenous relations are also defined for arrow diagram spaces. This timeone should pay attention to signs, so we make a full list (Fig.6). We denotethem by AP , AP ( n − , , AP ( n − , (or A T ) and AP ( n − , (or A T ). Theabove lemma explains why AP ( n − , is useless (it writes 0 = 0).9et us mention here the following two crucial points in the proofs of Theo-rems 2.5 and 3.6. Lemma 3.2.
For all n ≥ : Span(AP ( n − , ) ⊆ Span(AP ( n − , ) ∪ Span(AP ( n − , ) . Lemma 3.3.
Let A ∈ A and let X be a name among P , P ( n − , , P ( n − , , P ( n − , . Then A ∈ Span ⊥ (AX) ⇐⇒ S ( A ) ∈ Span ⊥ (X) . Defining the map d Applying Theorem 2.6, Lemma 3.1 and Lemma 3.3, gives immediately:
Lemma 3.4.
Let A ∈ A . Then the function h S ( A ) , I ( · ) i defines an invariantunder Reidemeister I and II moves for virtual knots with homology class K ifand only if A satisfies all the relations h A, AP i = 0 and D A, AP ( n − , E = 0 . This condition is easy to check with our naked eye, so we will be happy witha map d which can only detect invariance under Reidemeister III.Let A be an arrow diagram. We denote by • ( A ) ∈ A • the sum of all baseddiagrams that one can form by choosing a base arc in A . The map d is firstgoing to be defined on based diagrams. Definition 3.5.
We say that a based diagram B • is nice if the endpoints of itsbase arc belong to two different arrows.If B • is not nice, then we set d ( B • ) = 0.If B • is nice, then d ( B • ) is the degenerate diagram obtained from B • byshrinking the base to a point, multiplied by a sign ǫ ( B • ) defined as follows. Put η ( B • ) = (cid:26) +1 if the arrows that bound the base arc cross each other − , and let ↑ ( B • ) be the number of arrowheads at the boundary of the base arc.Then ǫ ( B • ) = η ( B • ) · ( − ↑ ( B • ) . The map d is extended linearly to A • .Finally, define: d ( A ) = d ( • ( A )) ∈ DA \∇ . (4)An example is shown on Fig.7. 10 a aba a+b aa+ba+bbb a+bbaa+b−Kbaa+b−Kba a+b−Kba+b−Ka AP 3 (n−2),2 or "A6T" aa+bba a+bb ba a+b−Ka+b−Kbaa a a a (n−3),3 or "A2T"
AP 1 (n−2),2 ++ ++ +/− K +/− Figure 6: The homogeneous arrow relations
Theorem 3.6 (Main Theorem) . Let A ∈ A satisfy the conditions of Lemma 3.4.Then the following are equivalent:1. A is an arrow diagram formula for invariants of virtual knots.2. d ( A ) = 0 modulo the triangle relations.3. A ∈ Span ⊥ ( A T ) . This theorem gives a formal proof to the fact that the h , i pairing enablesa uniformization of the formulas that depend on parameters: we shall see for11 a b−a + a b−a + + b bK+a−bK+a−ba b a b + + b =d a b a K+a−bb−a= mod
0 triangle relations
Figure 7: Boundary of the universal planar chain invariant for n = 2 + + aa a a + aaaa 1/2 Figure 8: An arrow polynomial in the kernel of d but not R -invariantinstance that the degenerate cases of Grishanov-Vassiliev’s invariants need notbe treated separately. See also Proposition 2 from [3], where points ( i ) and ( ii )are actually the same formula, or the special case 2 a = K in Theorem 3.10below. Note that the requirement of R invariance is necessary with the presentsettings – see Fig.8. Remark . It is possible to give a simplicial interpretation of the above map d , that makes it the first step towards a cohomology theory “`a la Vassiliev”[14, 15]. It will be done in a forthcoming paper. In [6], Grishanov-Vassiliev define an infinite family of arrow diagram formulasfor classical knots in M × R . Let us recall their construction. Definition 3.8. A naked arrow diagram is an arrow diagram with every deco-ration forgotten – as usual, up to oriented homeomorphism of the circle.A naked arrow diagram is called planar if no two of its arrows intersect.A chain presentation of such a diagram with n arrows is a way to numberits n + 1 internal regions from 1 to n + 1, in such a way that the numberingincreases when one goes from the left to the right of an arrow.Let U n be the sum of all planar isotopy equivalence classes of chain presen-tations of naked arrow diagrams with n arrows. U n is called the universal degree n planar chain ([6], Definition 1). 12or i = 1 , . . . , n + 1, let γ i ∈ Z \ { } . Given a chain presentation of a nakedplanar arrow diagram, and given such an ordered collection Γ, we construct anarrow diagram by decorating each arrow with the sum of the γ i ’s whose index i is located to the left of that arrow. The global decoration of the circle is set tobe the sum of all γ i ’s.The element of A n constructed that way from U n and Γ is denoted by Φ Γ ([6], Definition 2). Note that some of the summands in U n may lead to the sameelement of A n if some of the γ i ’s are equal; unlike Vassiliev-Grishanov, we donot forbid that.The arrow polynomial at the top of Fig.7 is the generic example for n = 2,with Γ = ( a, b − a, K − b ). Theorem 3.9.
For any ordered collection of non-zero homology classes
Γ =( γ , . . . , γ n +1 ) , the sum Φ Γ defined above enjoys the hypotheses of Theorem 3.6and thus defines an arrow diagram formula for virtual knots. This is an improvement of Theorem 1 from [6], since we remove the assump-tion that Γ is unambiguous ( i.e. here any of the γ i ’s may coincide), and weshow that Φ Γ is an invariant for virtual knots. In practice, Theorem 3.6 gives a very easy means of checking that an arrowpolynomial defines a virtual invariant. On another hand, finding virtual invari-ants when one has no clue of a potential formula demands to solve the systemof equations A T .We wrote a program to do this, and only a few results came, including thegeneralized Grishanov-Vassiliev’s planar chain invariants, and the following: Theorem 3.10.
Let K ∈ Z . The arrow polynomial of Fig.9 defines an invariantof virtual knots with homology class K for any parameter a ∈ Z \ { } . This seems to give a positive answer to T. Fiedler’s question about theexistence of N -invariants not contained in those from [3], Proposition 2.On the other hand, the sparse landscape of results leads to think that mostarrow diagram invariants might have infinite length – i.e. live in the algebraiccompletion of A , just like Fiedler’s N -invariants. a 0 + K−aa0 a + + a 0 Figure 9: An invariant of length 5 identically zero for closed braidsNotice that in case a = K , the formula has only 3 terms – but still definesan invariant. We compute it for the family of knots K i +1 drawn in Fig.10: I ( K i +1 ) = i ( i + 1) .
13t proves that there is no general algebraic formula expressing I in terms ofthe only invariants of degree 2 and finite length previously known (at least tothe author), namely Grishanov-Vassiliev’s length 3 invariant (Fig.7) – note thatthis invariant is already present in [2] for nullhomologous knots ( K = 0). K + + + + ++++ K Figure 10: K and K Recall the notations from section 2.2.
Lemma 4.1.
For all A ∈ A and G ∈ G , the following equality holds: hh A, G ii = h S ( A ) , I ( G ) i . The equality (( A, G )) = ( S ( A ) , I ( G )) holds for all G if and only if A has no symmetries other than the identity.Proof. Number the n arrows of A and fix a map σ : { , . . . , n } → {± } . Thegroup of symmetries of A σ , Aut( A σ ), identifies with a subgroup of Aut( A ).Also, the (abelian) group Aut( A ) naturally acts on the set {± } n . The orbitof σ is the set of maps σ such that A σ is equivalent to A σ under planarisotopies, and the cardinality of this orbit is the coefficient of A σ in the linearcombination S ( A ). By definition, the stabilizer of σ is the image of the injectivemap Aut( A σ ) ֒ → Aut( A ), whence, if we set O to be the set of orbits: S ( A ) = X [ σ ] ∈O | Aut( A ) || Aut( A σ ) | A σ . It follows that for any G , h S ( A ) , I ( G ) i = | Aut( A ) | P [ σ ] ∈O sign( σ )( A σ , I ( G ))= | Aut( A ) | (( A, G ))= hh A, G ii . G to be A σ . Then:( S ( A ) , I ( G )) = P [ σ ] ∈O sign( σ ) | Aut( A ) || Aut( A σ ) | ( A σ , I ( G ))= sign( σ ) | Aut( A ) || Aut( A σ ) | , while (( A, G )) = sign( σ ) . So one must have | Aut( A ) | = | Aut( A σ ) | for all σ , which can be true only if | Aut( A ) | = 1. Indeed, if ρ ∈ Aut( A ) \ { Id } , pick an arrow α of A such that ρ ( α ) = α and choose any σ such that σ ( α ) = 1 while σ ( ρ ( α )) = −
1. Necessarily ρ / ∈ Aut( A σ ), so that | Aut( A σ ) | < | Aut( A ) | . Proof of Lemma 3.2.
Figs.11 and 12 show eight A T relations, where it is as-sumed that the unseen parts are identical in all 48 diagrams (the dashed arrowsare dashed only for the sake of clarity). Up to P ( n − , , the combination(1) + (2) + 12 [(3) + (4) − (5) − (6) − (7) − (8)]gives the top A T relation shown on Fig.6. To get the other half of Span( A T ),just reverse the arrows in the previous equation, and change their markings from x to K − x . Proof of Lemma 2.3.
Let G d be the principal part of G . Since G is a Gaussdiagram formula it must satisfy the P relations (Theorem 2.6). Since G does nothave summands of degree higher than d , G d must satisfy the P ( n − , relations.Lemma 3.1 concludes the proof. Proof of Lemma 3.1.
Let A be an arrow diagram, and set G = S ( A ). By def-inition of S , any couple of Gauss diagrams that differ only by the sign of onearrow happen in G with opposite coefficients. S being linear, this implies: S ( A ) ⊂ M n ≥ Span ⊥ (P ( n − , ) . On the other hand, let G satisfy the P ( n − , equations. Define A = X ( G, A + ) · A , where the sum runs over all arrow diagrams, and the + operator decorates everyarrow with a + sign. It is easy to check that G = S ( A ).15 + ++ ab a+b (1) ++ a b a+b (2) a+baaa+baaaab aab ++ b bbb b b b bab ab a+baab a+b b a+b a+baaaa+bbb a+b b a+bab bab bb a a+baa+ba a ab ab aa+bb a ab (3) (4) a+ba+b Figure 11: Proof of Lemma 3.2 – part 1
Proof of Theorem 2.1.
By Theorem 2 . . moves from Fig.1 are the only Gauss pictures that canmatch Ω3 a . Since M. Polyak’s proof is local, it works in our framework as well.16 + ++ ab a+b a ++ bb ++ b a+b ba+ba+b a+ba+ba+ba a+ba+b aa+b b a+ba a aa+baa+ba a aba abaa+bb a+ba+ba+ba+ba+ba+ba aa+ba+bb b aa+bb b a+b abaa aaa+ba a aaa+b ab a+bb a+ba Figure 12: Proof of Lemma 3.2 – part 2
Proof of Theorem 2.5.
Let A ∈ A be an arrow diagram formula. It suffices toprove that the principal part of A , say A d , is an arrow diagram formula. ByTheorem 2.6, h S ( A ) , P i i = 0 for i = 1 , ,
3. Let us show that the same goes for A d :1 . P and S are homogeneous, so h S ( A d ) , P i = 0.17
31R R 2
Figure 13: Homological obstruction to R-moves2 . By Lemma 3.1: D S ( A d − ) , P ( d − , E = 0 (5) D S ( A d ) , P ( d − , E = 0 . (6)The equations h S ( A ) , P i = 0 together with 5 imply that D S ( A d ) , P ( d − , E = 0 . (7)Together with 6, we get: h S ( A d ) , P i = 0.3 . The last and crucial point: h S ( A ) , P i = 0 = ⇒ D S ( A d ) , P ( d − , E = 07 + Lemma . ⇒ D A d , AP ( d − , E = 0 and D A d , AP ( d − , E = 0 Lemma . ⇒ D A d , AP ( d − , E = 0 and D A d , AP ( d − , E = 0 Lemma . ⇒ D S ( A d ) , P ( d − , E D S ( A d ) , P ( d − , E = 0= ⇒ h S ( A d ) , P i = 0 . Proof of Theorem 3.6.
The proof will consist in defining and explaining the fol-lowing chain of equivalences. d ( A ) = 0 ⇔ ( • ( A ) , A T • ) = 0 ⇔ h A, A T i = 0 ⇔ h S ( A ) , I (R ) i = 0Notice that both extremities of this chain are homogeneous conditions (forthe right one, it follows from the proof of Theorem 2.5). So we may assume that A is homogeneous. 18 . d ( A ) = 0 ⇔ ( • ( A ) , A T • ) = 0.Let us call a degenerate diagram (with one degeneracy) monotonic if an ar-rowhead and an arrowtail meet at the degenerate point. The set of monotonicdiagrams forms a basis of DA / ∇ . It is clearly a generating set thanks to the ∇ relations, and it is free because every non monotonic diagram happens inexactly one relation, and every relation contains exactly one of them.We introduce the orthonormal scalar product ( , ) with respect to this basis.Let D be a monotonic degenerate diagram and B • a based diagram. It iseasy to check that the coordinate of d ( B • ) along D is given by( d ( B • ) , D ) = ( B • , A T • ( D )) , where A T • ( D ) is what we call the based D (seean example on Fig.14). b a A6T (D) b ab a+bb aa ++ a+b a+ba+ba b D Figure 14: The based 6-term relation associated with a degenerate diagram2 . ( • ( A ) , A T • ) = 0 ⇔ h A, A T i = 0.Let A and A ′ denote two arrow diagrams. We set:[ A, A ′ ] def = ( • ( A ) , A ′• )where A ′• is the based diagram obtained from A ′ by choosing any arc as a basearc. We have to show that this is a definition. If A = A ′ , then the right handside is unambiguously zero. If A = A ′ , then there are exactly | Aut( A ′ ) | sum-mands in • ( A ) that coincide with any fixed choice of base point in A ′ . So thepairing [ , ] is well-defined, and moreover it coincides with h , i .3 . h A, A T i = 0 ⇔ h S ( A ) , I (R ) i = 019y Theorem 2.6, h S ( A ) , I ( . ) i being invariant under R moves is equivalentto h S ( A ) , P i = 0 for any Polyak’s 8 T relation P . Since by hypothesis A ishomogeneous (say of degree n ), this is equivalent to S ( A ) actually satisfyingseparately the P ( n − , and the P ( n − , relations. Now apply successively Lem-mas 3.3 and 3.2 to terminate the proof. Proof of Theorem 3.9.
The fact that no γ i may be trivial gives immediately thecondition from 3.4. It is convenient here to check condition 3 of Theorem 3.6.In any A T relation, only three diagrams may have pairwise non intersectingarrows, and either all of these have, either no one has. The subsequent reducedrelations are shown on Fig.15 (the usual relations between the markings ofthe arrows have a natural equivalent in terms of Grishanov-Vassiliev’s regionmarkings). We say that a diagram with its regions marked is consistent if itsmarkings satisfy the chain presentation rule from Definition 3.8 – in other words,a diagram is consistent if it appears in Φ Γ . Consider the top relation of Fig.15,which can be written A − A − A . We see that:1. A is consistent if and only if A is consistent and i < j .2. A is consistent if and only if A is consistent and i > j .It follows that Φ Γ satisfies the A T relations. The proof for A T is similar. k ki ikj j ji A6T2A6T1 kj i j i ij kk Figure 15: The two kinds of reduced 6-term relations for planar diagrams
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