We present an easy example of mutant links with different Khovanov homology.
The existence of such an example is important because it shows that Khovanov
homology cannot be defined with a skein rule similar to the skein relation for
the Jones polynomial.Comment: 9 pages, 5 figure

Wehrli, Stephan M.

English

arXiv

We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial

Wehrli, Stephan

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Syracuse University SURFACE Mathematics - Faculty Scholarship Mathematics 1-27-2003 Khovanov Homology and Conway Mutation Stephan Wehrli Syracuse University Follow this and additional works at: https://surface.syr.edu/mat  Part of the Mathematics Commons Recommended Citation Wehrli, Stephan, "Khovanov Homology and Conway Mutation" (2003). Mathematics - Faculty Scholarship. 112. https://surface.syr.edu/mat/112 This Article is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics - Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:math/0301312v1  [math.GT]  27 Jan 2003Khovanov Homology and ConwayMutationStephan Wehrli, Universität Basel, January 2003AbstractIn this article, we present an easy example of mutant links withdifferent Khovanov homology. The existence of such an example isimportant because it shows that Khovanov homology cannot be definedwith a skein rule similar to the skein relation for the Jones polynomial.1 IntroductionIn [Kh] M. Khovanov assigned to the diagram D of an oriented link L abigraded chain complex C∗,∗(D), with a differential d that maps the chaingroup Ci,j(D) into Ci+1,j(D). He proved that the homotopy equivalenceclass of graded chain complex C∗,∗(D) only depends on the oriented link L.In particular, the homology groups Hi,j(D) (considered up to isomorphism)and the graded Poincaré polynomialKh(L)(t, q) :=∑i,jtiqj dimQ(Hi,j(D) ⊗ Q) ∈ Z[t, t−1, q, q−1]are link invariants. The aim of this paper is to give an example of orientedmutant links which are separated by the polynomial Kh and to prove thatconsequently the invariant Kh does not satisfy a skein relation similar tothe skein relation for the Jones polynomial.2 Skein equivalenceIn this section we briefly recall the definition of skein equivalence givenin [Ka]. A triple (L+, L−, L0) of oriented links is called a skein triple, if theoriented links L+, L− and L0 possess diagrams which are mutually identicalexcept in a small neighborhood, where they are respectively consistent with, and .Definition 1 The skein equivalence is the minimal (with respect to set-theoretical inclusion) equivalence relation ”∼” on the set of oriented linkssuch that1. L ∼ L′ when L and L′ are isotopic,2. L0 ∼ L′0 and L− ∼ L′−imply L+ ∼ L′+,3. L0 ∼ L′0 and L+ ∼ L′+ imply L− ∼ L′−,1for any two skein triples (L+, L−, L0) and (L′+, L′−, L′0).It is easy to see that such a minimal relation as postulated in the definitionactually exists. The definition is motivated by the following: Assume we aregiven an invariant f of oriented links, such as the Jones polynomial, whichtakes values in an arbitrary ring R and satisfies a relationαf(L+) + βf(L−) + γf(L0) = 0,where α, β ∈ R∗ and γ ∈ R. Then f(L+) is determined by f(L0) andf(L−), and f(L−) is determined by f(L0) and f(L+). The minimality of”∼” implies:Theorem 1 Let L and L′ be skein equivalent. Then f(L) = f(L′).3 Conway mutationThe mutation of links was originally defined in [Co]. We will use the defini-tion given in [Mu]. In Figure 1, the rectangular boxes represent an oriented2-tangle T . Let h1, h2 and h3 be the half-turns about the indicated axes.T TThh1 3h2Figure 1: The half-turns h1, h2 and h3Define three involutions ρ1, ρ2 and ρ3 on the set of oriented 2-tangles byρ1T := h1(T ), ρ2T := −h2(T ) and ρ3T := −h3(T ) (where −h2(T ) and−h3(T ) are the oriented 2-tangles obtained from h2(T ) and h3(T ) by re-T T1 2Figure 2: The closure of the composition of T1 and T2versing the orientations of all strings). For two oriented 2-tangles T1 andT2, denote by T1T2 the composition of T1 and T2 and by (T1T2)∧ the closureof T1T2 (see Figure 2).2Definition 2 Two oriented links L and L′ are called Conway mutants ifthere are two oriented 2-tangles T1 and T2 such that for an involution ρi(i = 1, 2, 3) the links L and L′ are respectively isotopic to (T1T2)∧ and(T1ρiT2)∧.Theorem 2 Let L and L′ be Conway mutants. Then L and L′ are skeinequivalent.Proof. The proof goes by induction on the number n of crossings of T2. Forn ≤ 1, T2 and ρiT2 are isotopic, whence L ∼ L′. For n > 1, modify a crossingof T2 to obtain a skein triple of tangles (T+, T−, T0) (with either T+ = T2or T− = T2, depending on whether the crossing is positive or negative).Denote by (L+, L−, L0) and (L′+, L′−, L′0) the skein triples corresponding to(T+, T−, T0) and (ρiT+, ρiT−, ρiT0) respectively (i.e. L+ = (T1T+)∧, L− =(T1T−)∧ and so on). By induction, L0 ∼ L′0. Therefore, by the definitionof skein equivalence, L+ ∼ L′+ if and only if L− ∼ L′−. In other words,switching a crossing of T2 does not affect the truth or falsity of the assertion.Since T2 can be untied by switching crossings, we are back in the case n ≤ 1.4 Mutant links with different Khovanov homologyLet V (L)(q) := Kh(L)(−1, q) denote the graded Euler characteristic of C(D)and W (L)(t) := Kh(L)(1, q) the ordinary (ungraded) Poincaré polynomial.As is shown in [Kh], V is just an unnormalized version of the Jones polyno-mial. By the results of sections 2 and 3, the Jones polynomial is invariantunder Conway mutation. On the other hand, the following theorem givesan example of mutant links which are separated by W .Theorem 3 Let Ki (i = 1, 2) be a (2, ni) torus link, ni > 2. Then theoriented linksL := ⊔ (K1♯K2) and L′ := K1 ⊔ K2are Conway mutants with different W polynomial. Here, denotes thetrivial knot and K1♯K2 is the connected sum of the oriented links K1 and K2.Note that the connected sum is well-defined even if Ki has two components,because in this case the link Ki is symmetric in its components.Proof. From Figure 3 it is apparent that L and L′ are Conway mutants.The Khovanov complex of the trivial knot is. . . −→ 0 −→ 0 −→ A −→ 0 −→ 0 −→ . . .3K K2T T121T T1 2K K1 2Lh’L1Figure 3: L and L′ are Conway mutantsand rank(A) = 2, whence W ( ) = 2. By [Kh, Proposition 33], Kh ismultiplicative under disjoint union, and so W (L) = 2W (K1♯K2). On theother hand, by [Kh, Proposition 35],W (Ki) = 2 + t−2 + t−3 + . . . + t−(ni−1) + t−niif ni is odd andW (Ki) = 2 + t−2 + t−3 + . . . + t−(ni−1) + 2t−niif ni is even. Since ni > 2, W (Ki) is not divisible by 2. But then W (L′) =W (K1)W (K2) is not divisible by 2 and hence W (L′) 6= W (L). Theorems 1, 2 and 3 immediatly imply:Corollary 1 The W polynomial does not satisfy a relation of the kind men-tioned in section 2.Remark. Theorem 3 remains true if we also allow torus links Ki with ni < −2(this may be seen using [Kh, Corollary 11], which relates the Khovanovhomology of a link to the Khovanov homology of its mirror image). Thecondition |ni| > 2 is necessary. In fact, if one of the |ni| is ≤ 1, then thecorresponding torus link Ki is trivial and hence L and L′ are isotopic. Ifn2 = 2, then L and L′ look as is shown in Figure 4. Note that both L − L0and L′ − L′0 are isotopic to the link ⊔ K1. Using [Kh, Corollary 10], onecan show that both Hi,j(L) and Hi,j(L′) are isomorphic to Hi+2,j+5( ⊔K1) ⊕Hi,j+1( ⊔ K1). The cases n2 = −2 and n1 = ±2 are similar.LK1 K1’L’L 00LFigure 4: L and L′ for the case n2 = 245 Computer Calculations with KHOHOTables 1 and 2 show the Khovanov homology of L and L′ for the casen1 = n2 = 3. The tables where generated using A. Shumakovitch’s programKhoHo [Sh]. The entry in the i-th column and the j-th row looks likea[b]c,where a is the rank of the homology group Hi,j, b the number of factorsZ/2Z in the decomposition of Hi,j into p-subgroups, and c the rank of thechain group Ci,j. The numbers above the horizontal arrows denote the ranksof the chain differentials.In the examples, only 2-torsion occurs. It has been conjectured by A. Shu-makovitch that this is actually true for arbitrary links. The reader mayverify that not only the dimensions but also the torsion parts of the Hi,j aredifferent for L and L′.The dimensions of the Ci,j agree because there is a natural one-to-one cor-respondence between the Kauffman states of L and L′ (which re-proves thefact that the Jones polynomial is invariant under Conway mutation).We do not know the answer to the following question:Question: Does there exist a pair of mutant oriented knots with distinctKhovanov homology?According to the database of A. Shumakovitch, no such pair of knots with13 or less crossings exists. In particular, the Kinoshita-Terasaka knot andthe Conway knot (the knots depicted in Figure 5) are mutant knots withthe same Khovanov homology (see Table 3).Figure 5: The Kinoshita-Terasaka knot and the Conway knotReferences[Ba] Dror Bar-Natan, On Khovanov’s Categorification of the Jones poly-nomial, Algebraic and Geometric Topology 2-16, 337-370, 2002,arXiv:math.QA/0201043.[Co] John H. Conway, An enumeration of knots and links and some oftheir related properties, Computational problems in Abstract Algebra,Pergamon Press, New York, 329-358, 1970.5[Ja] Magnus Jacobsson, An invariant of link cobordisms from Khovanov’shomology theory, arXiv:math.GT/0206303v1.[Ka] Akio Kawauchi, A Survey of Knot Theory, Birkhäuser, 1996.[Kh] Mikhail Khovanov, A categorification of the Jones polynomial, DukeMath. J. 101(3), 359-426, 1999, arXiv:math.QA/9908171.[Mu] J. Murakami, The Parallel Version of Polynomial Invariants of Links,Osaka J. Math. 26, 1-55, 1989.[Sh] Alexander Shumakovitch, KhoHo, Basel 2002, see http://www.geometrie.ch/KhoHo.[Vi] Oleg Viro, Remarks on the definition of Khovanov Homology,arXiv:math.GT/0202199.6-6 -5 -4 -3 -2 -1 0-2 112 4 2-4 02−−→ 06−−→ 06−−→ 241 5 10 18 13 5-6 01−−→ 06−−→ 015−−→ 028−−→ 233−−→ 018−−→ 166 24 36 38 14 4-8 06−−→ 030−−→ 060−−→ 074−−→2[2]54−−→ 018−−→ 0415 45 44 28 5 1-10 015−−→ 060−−→ 190−−→ 274−−→0[2]33−−→ 06−−→ 0120 39 20 6-12 020−−→ 160−−→1[1]60−−→ 228−−→ 0615 13 2-14 015−−→2[1]30−−→0[1]15−−→ 025-16 16−−→1[1]6-18 11Table 1: Ranks of Hi,j and Ci,j and ranks of the chain differentials for thedisjoint union of the unknot and the granny-knot7-6 -5 -4 -3 -2 -1 0-2 112 4 2-4 02−−→ 06−−→ 06−−→ 241 5 10 18 13 5-6 01−−→ 06−−→ 015−−→ 028−−→ 233−−→ 018−−→ 166 24 36 38 14 4-8 06−−→ 030−−→ 060−−→ 074−−→2[2]54−−→ 018−−→ 0415 45 44 28 5 1-10 015−−→ 060−−→ 190−−→ 274−−→0[2]33−−→ 06−−→ 0120 40 20 6-12 020−−→ 060−−→0[2]60−−→ 228−−→ 0615 13 2-14 015−−→2[1]30−−→0[1]15−−→ 026-16 06−−→0[2]6-18 11Table 2: Ranks of Hi,j and Ci,j and ranks of the chain differentials for thedisjoint union of two trefoil knots8-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 61 5 10 10 6 1901−−−→06−−−→015−−−→020−−→016−−→18−−→011 12 55 125 154 107 41 4701−−−→013−−−−→067−−−→0180−−−→0279−−−→0261−−→1149−−→0[1]45−−→045 55 260 658 951 792 371 89 6505−−−→060−−−→0315−−−−→0918−−−→01609−−−→01743−−−→11164−−→1[1]461−−→095−−→0610 126 663 1895 3159 3104 1767 565 91 43010−−−→0136−−−→0789−−−→02558−−−−→05054−−−→16264−−−→24873−−−→1[1]2333−−→0656−−→095−−→0410 158 983 3241 6238 7166 4871 1916 417 44 11010−−→0168−−−→01141−−−→04224−−−→09479−−−−→213406−−−→1[1]12038−−−→1[2]6788−−−→02333−−→0461−−→045−−→016 113 859 3375 7607 10219 8186 3850 1023 141 8-106−−→0119−−→0972−−−→04234−−−→010982−−−→117827−−−−→3[1]18408−−−→2[1]12038−−−→04873−−−→01164−−→0149−−→081 42 432 2128 5788 9186 8639 4766 1498 245 16-301−−→043−−→0474−−→02560−−−→07916−−−→214976−−−→2[1]17827−−−−→1[1]13406−−−→06264−−−→01743−−−→0261−−→0165 113 784 2708 5207 5774 3704 1350 259 20-505−−→0118−−→0897−−→03492−−−→17916−−−→1[2]10982−−−→1[1]9479−−−−→05054−−−→01609−−−→0279−−−→02010 152 745 1814 2418 1805 753 165 15-7010−−→0162−−→0897−−→12560−−−→2[1]4234−−−→14224−−−→02558−−−−→0918−−−→0180−−−→01510 108 365 606 535 254 61 6-9010−−→0118−−→1474−−→1[1]972−−−→01141−−−→0789−−−→0315−−−−→067−−−→065 38 80 88 48 12 1-1105−−→043−−→1[1]119−−→0168−−−→0136−−−→060−−−→013−−−−→011 4 6 4 1-1301−−→16−−→010−−→010−−−→05−−−→01Table 3: Ranks of Hi,j and Ci,j and ranks of the chain differentials for either the Kinoshita-Terasaka knot or the Conwayknot (both have the same Khovanov homology)9

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