Oriented Local Moves and Divisibility of the Jones Polynomial
aa r X i v : . [ m a t h . G T ] M a r Oriented Local Moves andDivisibility of the Jones Polynomial
Paul Drube
Department of Mathematics & StatisticsValparaiso University [email protected]
Puttipong Pongtanapaisan
Department of MathematicsUniversity of Iowa [email protected]
Abstract
For any virtual link L = S ∪ T that may be decomposed into a pair of oriented n -tangles S and T , an oriented local move of type T T ′ is a replacement of T with the n -tangle T ′ in a way that preserves the orientation of L . After developing ageneral decomposition for the Jones polynomial of the virtual link L = S ∪ T in termsof various (modified) closures of T , we analyze the Jones polynomials of virtual links L , L that differ via a local move of type T T ′ . Succinct divisibility conditions on V ( L ) − V ( L ) are derived for broad classes of local moves that include the ∆-moveand the double-∆-move as special cases. As a consequence of our divisibility result forthe double-∆-move, we introduce a necessary condition for any pair of classical knotsto be S -equivalent. For any link L , the Jones polynomial V ( L ) ∈ Z [ t / , t − / ] is a Laurent polynomial in thevariable t / . After being introduced by Jones himself [3], the Jones polynomial was recastby Kauffman in terms of his bracket polynomial [5] . For any unoriented link diagram L ,the bracket polynomial h L i ∈ Z [ A, A − ] is an invariant of framed links that may be definedrecursively via the local relations shown below. h i = A h i + A − h ih ∪ L i = ( − A − A − ) h L ih i = 11or an oriented link diagram with the bracket polynomial h L i , one may obtain the Jonespolynomial of the associated link by evaluating f ( L ) = ( − A ) − w ( L ) h L i at A = t − / , where w ( L ) is the writhe of L . We henceforth refer to the intermediate polynomial f ( L ) as theauxiliary polynomial of L .The rest of this paper assumes a basic familiarity with the Jones polynomial and theKauffman bracket. For more information on these topics, see Kauffman [5] or Lickorish [8].The Jones polynomial was subsequently generalized to virtual links by Kauffman [6]. Theresulting virtual link invariant, sometimes referred to as the Jones-Kauffman polynomial,may be defined in terms of the Kauffman bracket using the same local relations as aboveand the same evaluation of f ( L ) = ( − A ) − w ( L ) h L i at A = t − / . For a full discussion ofvirtual links and their topological importance, consult the surveys [6, 7].Now consider the unoriented virtual link diagram D , and suppose that D = S ∪ T may bedecomposed into the pair of n -tangles S and T . An (unoriented) local move of type T T ′ is a replacement of T with the n -tangle T ′ while leaving S unchanged, transforming D into adiagram D ′ = S ∪ T ′ of some (possibly distinct) virtual link. Local moves include operationsas ubiquitous as the simple crossing change (on 2-tangles), the ∆-move (on 3-tangles), andthe so-called forbidden moves of virtual links (on 3-tangles). An oriented local move of type T T ′ is a replacement of the oriented n -tangle T with the oriented n -tangle T ′ in a waythat preserves the orientation of all endpoints of T .The primary goal of this paper is to investigate how the auxiliary polynomial of a virtuallink behaves under a variety of oriented local moves. In particular, we consider any pairof oriented links L , L that differ via a finite sequence of some fixed move, and developdivisibility conditions for the auxiliary polynomial f ( L ) − f ( L ). This places a necessarycondition upon whether a given pair of links may be connected via repeated application ofa particular local move and, in the case where L is a knot and L is unknot, may be usedto show that the move in question is not an unknotting move.Divisibility conditions of the type above date back to Jones [3], who showed f ( K ) − f ( K )is divisible by A − A − A + 1 for any pair of classical knots K , K . Our methods moreclosely follow that of Ganzell [2], who used the bracket polynomial to find divisibility condi-tions for f ( K ) − f ( K ) when K , K were a pair of knots connected by various (unoriented)local moves. Ganzell showed that f ( K ) − f ( K ) is divisible by A − f ( K ) − f ( K ) is divisible by A − A − A + 1for any pair of classical knots that differ by a ∆-move, and that f ( K ) − f ( K ) is divisibleby A − A − A + 1 for any pair of virtual knots that differ by a forbidden move. For addi-tional results of a similar type see Nikkuni [12], who showed that f ( L ) − f ( L ) is divisibleby ( A − − n ( A − + A − + 1)( A − + 1) for any pair of oriented classical links L , L thatdiffer by a C n move (for every n ≥ T T ′ , and for some collection of links S consider all pairs L , L ∈ S that are related via a finite sequence of moves of fixed type T T ′ . We saythat p ( A ) ∈ Z [ A, A − ] is a maximal divisor for S with respect to T T ′ if, whenever q ( A ) ∈ Z [ A, A − ] divides every polynomial of the form f ( L ) − f ( L ), then q ( A ) divides p ( A ). Note that one may immediately conclude that p ( A ) is a maximal divisor if there exist Throughout this paper, we use a generalized notion of tangle that allows for closed components withoutendpoints on the boundary. , L ∈ S such that f ( L ) − f ( L ) = p ( A ). Such links have been found for every divisormentioned in the previous paragraph, proving their maximality within the stated collectionof links [3, 2, 12].Our results differ from those of Ganzell [2] in that all of our local moves are oriented. Thisnarrows the classes of links that may be connected via repeated application of a given move,and our divisibility conditions for f ( L ) − f ( L ) need not extend to any pair of links thatdiffer via an unoriented version of the same move. On the other hand, dealing with orientedmoves allows us to more easily tackle local moves with a large number of outgoing strands.Observe that a (maximal) divisor for some unoriented local move T T ′ may be obtainedby separately determining a (maximal) divisor for every orienation that is compatible withboth T and T ′ , and then taking the greatest common divisor of those polynomials. This paper is organized as follows. In Section 2 we introduce our general technique fordecomposing the auxiliary polynomial of an arbitrary virtual link of the form T ∪ T ′ . Theorem2.3 gives f ( T ∪ T ′ ) = P m ∈P n q m f ( e T B ( m )), where the e T B ( m ) represent the various closuresof T (via every 2-equal matching m in P n ) and q m ∈ Z [ A, A − ] are unspecified Laurentpolynomials that depend upon the structure of T ′ .In Section 3, we apply Theorem 2.3 to find maximal divisors for a variety of orientedlocal moves. Subsection 3.1 focuses upon local moves that involve rotation of a classical n -tangle by a fixed number of strands. Subsections 3.2 and 3.3 present lengthier treatmentsfor a pair of local moves that do not conform to the methods of Subsection 3.1, namelythe double-∆-move for classical 6-tangles and a rotational move for virtual 2-tangles. Inthe case of the double-∆-move this prompts an intriguing new result on S-equivalence ofknots, with Corollary 3.12 stating that two classical knots K , K may be S-equivalent onlyif f ( K ) − f ( K ) is divisible by A − A + A − A − A + A − A + 1. For any n ≥
1, consider the set [2 n ] = { , , . . . , n } . A 2-equal partition of [2 n ] is a partitionof [2 n ] into n disjoint sets of size 2. Every 2-equal partition P may be associated with a(2-equal) matching on the circle, in which the element i ∈ [2 n ] corresponds to the pointalong the unit circle with radial coordinate θ = − πin , and the points corresponding to i and j are connected via an arc within the unit circle if and only if i and j belong to the sameblock of P . We denote the set of all such matchings on 2 n points by P n .An element of P n is said to be noncrossing if it may be drawn so that no two arcs intersect.We denote the set of all noncrossing (2-equal) matchings on 2 n points by M n . It is wellknown that |M n | = n +1 (cid:0) nn (cid:1) , the n th Catalan number. We henceforth refer to any matchingvia the blocks of the associated partition. See Figure 1 for basic examples.In all that follows, we assume that matchings have been drawn such that no two arcsintersect more than once and no three arcs have a common intersection. Given these condi-tions, there exists an obvious bijection between P n and the set of unoriented virtual n -tangleswith zero classical crossings. For any m ∈ P n , a diagram of the associated tangle T m may3 Figure 1: The three elements m = ((1 , , (3 , m = ((1 , , (2 , m = ((1 , , (2 , P , among which m and m also belong to M be obtained by replacing all intersections in m with virtual crossings and interpreting theunit circle as the tangle boundary. When referring to T m , we will always take a diagram inwhich the endpoint corresponding to i has radial coordinate θ = − πin .Now take any virtual n -tangle T . Our formalism involving 2-equal matchings is motivatedby the fact that every Kauffman state of T is isotopic to T m for some m ∈ P n . Gatheringterms from the Kauffman state sum that resolve to the same T m , this implies that h T i de-composes as h T i = P m ∈P n p m h T m i , where each p m ∈ Z [ A, A − ] is a Laurent polynomial thatdepends upon the structure of T . If our tangle T lacks virtual crossings, this decompositionclearly reduces to h T i = P m ∈M n p m h T m i .Directly pertinent to this paper is the situation where a virtual link L may be decomposedinto the two n -tangles T and T ′ . In this case, we always take a diagram of L in which T appears as described above, and then order the endpoints of T ′ so that the i th endpoint of T ′ is identified with the i th endpoint of T . Here we adopt the shorthand L = T ∪ T ′ .For L = T ∪ T ′ , observe that smoothing every crossing in T ′ (while leaving T unchanged)produces a virtual link T ( m ) = T ∪ T ′ m for some m ∈ P n . We refer to this link as the closureof T by m . Diagrammatically, note that T ( m ) may be obtained from T by inverting all arcsof m across the unit circle, replacing all intersections in the resulting matching with virtualcrossings, and attaching the i th endpoint of m to the i th external strand of T .See Figure 2 for an illustration of every closure for an arbitrary 2-tangle T . In theparticular case of a 2-tangle, notice that the two closures without virtual crossings correspondto the numerator closure and denominator closure of T . T
12 3 4 T ( m ) T
12 3 4 T ( m ) T
12 3 4 T ( m ) Figure 2: The three closures of the 2-tangle T , corresponding to the matchings m =((1 , , (3 , m = ((1 , , (2 , m = ((1 , , (2 , h T i may be written in terms of the h T m i , the bracket polynomial forthe link h T ∪ T ′ i may be written in terms of the h T ( m ) i . See Fish and Keyman [1] for adistinct derivation of a result equivalent to Proposition 2.1.4 roposition 2.1. Let L = T ∪ T ′ be any virtual link that has been decomposed into a pairof n -tangles T and T ′ . Then h T ∪ T ′ i = X m ∈P n q m h T ( m ) i where the q m ∈ Z [ A, A − ] are Laurent polynomials that depend upon the structure of T ′ . We wish to translate Proposition 2.1 into a result involving the auxiliary polynomial f ( T ∪ T ′ ). The difficulty is that this must be done in a way that doesn’t require internalknowledge of T . In particular, once we declare a specific orientation for L = T ∪ T ′ , manyclosures T ( m ) may fail to be compatible with that orientation. Simply defining f ( T ( m )) inthose cases would require a reorientation of some proper subset of the strands from T , anaction whose effect on the writhe may require internal knowledge of T .One way of avoiding this problem is to work with oriented and disoriented resolutions ata real crossing, so that the Kauffman states are purely virtual magnetic graphs. See Kamadaand Miyazawa [4] and Miyazawa [9] for results involving the resulting generalization of theKauffman-Jones polynomial. Unfortunately, working with purely virtual magnetic graphssignificantly complicates much of what follows, and we instead use the Kauffman skeinrelation to systematically replace all problematic closures with diagrams that respect thedesired orientation.So consider any word ~v of length 2 n that features exactly n instances of + and n instancesof − , and let v i denote the i th letter of ~v . We say that the 2 n -tangle T has orientation ~v if its i th endpoint has an outbound orientation precisely when v i = +. For any tangle T oforientation ~v , we construct a braid B ~v on 2 n strands as follows:1. Identify the longest initial subword s of ~v that is of the form (+ − ) k or (+ − ) k +.2. If | s | = 2 n , terminate the procedure. If | s | < n , identify the smallest index j > | s | such that v j = v | s | +1 and add σ j − σ j − . . . σ | s | +1 to the end of B ~v .3. Define ~v ′ to be the length 2 n word whose letters satisfy v ′| s | +1 = v j , v ′ i = v i − for | s | + 2 ≤ i ≤ j , and v ′ i = v i otherwise. Then return to Step ~v = ~v ′ .As the new word ~v ′ in Step ~v , the procedure above terminates after a finite number of steps. See Figure 3 forthe braids B ~v associated with each (fundamentally distinct) orientation ~v on 4 or 6 endpoints.Now take the virtual link L = T ∪ T ′ , and assume that L has been oriented in such away that T has orientation ~v . Then identify the i th endpoint of T with the bottom of the i th strand of B ~v . This produces an oriented n -tangle T B whose endpoints alternate betweeninbound and outbound strands, a situation that we henceforth refer to as the “standardorientation” for an n -tangle. For any closure T ( m ) of T , there exists an associated closure T B ( m ) of T B that is produced by attaching T ( m ) − T to the endpoints of T B .As one final modification to ensure that our closures respect the orientation ~v , we trans-form each closure T B ( m ) into e T B ( m ) by replacing the neighborhood of every virtual crossingin T B ( m ) − T B as shown in Figure 4. The labels on the left side of that figure indicate5 - + -+ - + - B ~v = id + + - -+ - + - B ~v = σ + - + - + -+ - + - + - B ~v = id + + - - + -+ - + - + - B ~v = σ + + - + - -+ - + - + - B ~v = σ σ + + + - - -+ - + - + - B ~v = σ σ σ Figure 3: Up to cyclic permutation, the distinct orientation braids B ~v on 4 and 6 endpoints. a a a a ⇒ Figure 4: Replacing the neighborhood of a virtual crossing in T B ( m ) − T B .the endpoints of T B to which each strand is eventually attached, where we assume that a < a < a < a to ensure that the local operation is well-defined.Lemma 2.2 shows that the original closures T ( m ) from the decomposition of Proposition2.1 may be swapped out for the modified closures e T B ( m ): Lemma 2.2.
Let T be an n -tangle with orientation ~v , and take any m ∈ P n . Then thereexist p µ ∈ Z [ A, A − ] such that h T ( m ) i = X µ ∈P n p µ h e T B ( µ ) i Proof.
For any m ∈ P n , we first find q µ ∈ Z [ A, A − ] such that h T ( m ) i = P µ ∈P n q µ h T B ( µ ) i .For any m ∈ P n , we then provide e q µ ∈ Z [ A, A − ] such that h T B ( m ) i = P µ ∈P n e q µ h e T B ( µ ) i .So assume B ~v = σ i σ i . . . σ i M , and let b k = σ i σ i . . . σ i k be the initial subword of B ~v oflength k . For each 0 ≤ k ≤ M , we define T b k to be the n -tangle created by identifying the i th endpoint of T with the bottom of the i th strand of b k , so that T b = T and T b M = T B .For fixed 1 ≤ k ≤ M and any m ∈ P n we demonstrate there exist q µ ∈ Z [ A, A − ] such that h T b k − ( m ) i = P µ ∈P n q µ h T b k ( µ ) i .Take T b k ( m ), and consider the neighborhood of the final crossing σ k from b k in T b k ( m ),located just inside the boundary of T b k . The Kauffman-Jones skein relation gives the fol-lowing, where the horizontal line denotes the external boundary of T b k and the − A ± termis determined by the writhe of the nugatory crossing introduced on the right side. h i = A h i + A − h i = A ( − A ± ) h i + A − h i h i = A h i − A ( − A ± ) h i Notice that the first term in the second equation is simply h T b k − ( m ) i . Since the diagramassociated with the final term of the second equation lacks classical crossings away from T b k ,after the removal of trivial split components and nugatory crossings it must be equivalent to T b k ( µ ) for some µ ∈ P n . Thus h T b k − ( m ) i = A h T b k ( m ) i − A ( − A − − A ) t ( − A ) t h T b k ( µ ) i for some t ≥ t ∈ Z , and µ ∈ P n . Repeatedly applying this result until reaching k = M allows us to conclude that h T ( m ) i = P µ ∈P n q µ h T B ( µ ) i for some q µ ∈ P n .Now consider the set of closures { T B ( m ) } m ∈P n , and let S k denote the subset of those linksthat feature precisely k virtual crossings away from T . We induct on k ≥
0, showing that any T B ( m ) ∈ S k may be written as h T B ( m ) i = P µ ∈P n p m,µ h e T B ( µ ) i for some p m,µ ∈ Z [ A, A − ].The case of k = 0 follows from the fact that T B ( m ) = e T B ( m ) for any closure that lacksvirtual crossings away from T B . So take any T B ( m ) ∈ S k , where k ≥
1, and consider theassociated link e T B ( m ). In the neighborhood of any classical crossing in e T B ( m ) − T B , theKauffman-Jones skein relation gives h i = A h i + A − h i = A h i + A − h i Resolving every classical crossing of e T B ( m ) − T B as above gives h e T B ( m ) i = A k h T B ( m ) i + P α q α h D α i for q α ∈ Z [ A, A − ] and some collection of link diagrams D α , each of which contain T B , lack classical crossings away from T B , and have at most k − T B . Up to trivial split components and nugatory crossings, each D α is then equivalentto some closure T B ( m α ) that has at most k − T B . It followsthat h e T B ( m ) i = A k h T B ( m ) i + P m ∈P n p ′ m h T B ( m α ) i for some p ′ m ∈ Z [ A, A − ] and some set ofclosures T B ( m α ) that each contain at most k − T B . Rearranginggives h T B ( m ) i = A − k h e T B ( m ) i − A − k P m ∈P n p ′ m h T B ( m α ) i for some set of closures T B ( m α )that each contain at most k − T B . Applying the inductiveassumption allows us to conclude h T B ( m ) i = P µ ∈P n e q µ h e T B ( µ ) i for some e q µ ∈ Z [ A, A − ].Pause to note that Lemma 2.2 is dependent upon the specific algorithm by which wetransformed each T ( m ) into e T B ( m ). That algorithm is certainly only one of many waysto systematically replace all closures with counterparts that are compatible with the givenorientation. It is an open question as to whether the divisors derived in Section 3 are identicalto those that would result from a different definition of e T B ( m ).We are now ready for the primary theorem of this section, which translates the decom-position of h T ∪ T ′ i from Proposition 2.1 to a decomposition of the auxiliary polynomial f ( T ∪ T ′ ), no matter the orientation on T ∪ T ′ . Theorem 2.3.
Let L = T ∪ T ′ be an oriented virtual link that has been decomposed into the n -tangles T and T ′ . Then f ( T ∪ T ′ ) = X m ∈P n q m f ( e T B ( m )) where the q m ∈ Z [ A, A − ] are Laurent polynomials that depend upon the structure of T ′ . roof. Applying Lemma 2.2 to Proposition 2.1, we immediately know that h T ∪ T ′ i = P m ∈P n p m h e T B ( m ) i for some p m ∈ Z [ A, A − ]. In order to translate this result to f ( T ∪ T ′ ),we need to show that e T B ( m ) respects the orientation on L for every m ∈ P n .Begin by observing that, no matter the original orientation on L , the modified tangle T B always has endpoints that alternate between inward and outward strands. As T B hasstandard orientation, it is straightforward to show that a particular closure T B ( m ) respectsthe orientation on T B if and only if m is non-crossing. Thus e T B ( m ) = T B ( m ) respects thegiven orientation for all m ∈ M n . For matchings m ∈ P n with at least one crossing, considerthe virtual link L m that may be obtained from e T B ( m ) by replacing the neighborhood of everyvirtual crossing in e T B ( m ) as shown below. ⇒ As each L m lacks crossings away from T B , up to trivial split components it is equivalentto T B ( m ) for some m ∈ M n . It follows that L m respects the orientation on T B . As the localmove shown above is always orientation-preserving, we may conclude that e T B ( m ) respectsthe given orientation for any m ∈ P n − M n .Now assume that our original n -tangles have writhes w ( T ) = w and w ( T ′ ) = w ′ . Wethen have h T ∪ T ′ i = ( − A ) w + w ′ f ( T ∪ T ′ ). Knowing that every modified closure e T B ( m ) iscompatible with the orientation on T ∪ T ′ , we also have h e T B ( m ) i = ( − A ) w + e w m f ( e T B ( m ))for every m ∈ P m , where e w m is dependent upon the structure of e T B ( m ) − T . The theoremfollows by substituting these results into h T ∪ T ′ i = P m ∈P n p m h e T B ( m ) i .See Figure 5 for an illustration of the modified closures e T B ( m ) from Theorem 2.3, when T is a 2-tangle with either of the orientations from Figure 3.If the original link L = T ∪ T ′ lacks virtual crossings, observe that the closures T B ( m )involving m ∈ P n − M n never contribute to the summation of Theorem 2.3. Since we alsohave e T B ( m ) = T B ( m ) for every m ∈ M n , we draw the following corollary. Corollary 2.4.
Let L = T ∪ T ′ be an oriented classical link that has been decomposed intothe n -tangles T and T ′ . Then f ( T ∪ T ′ ) = X m ∈M n q m f ( T B ( m )) where the q m ∈ Z [ A, A − ] are Laurent polynomials that depend upon the structure of T ′ . As our primary application of Theorem 2.3, we investigate how various local moves effectthe auxiliary polynomial of a virtual link. So let T be an n -tangle with orientation ~v , andlet φ be a local move that replaces T with another n -tangle φ ( T ) = T of orientation ~v .Then consider the virtual links L = T ∪ T ′ and φ ( L ) = T ∪ T ′ , where T ′ is an arbitrary8
12 3 4 T
12 3 4 T
12 3 4 T
12 3 4 T
12 3 4 = T
12 3 4 T
12 3 4 = T
12 3 4
Figure 5: The three modified closures e T B ( m ) for a 2-tangle with orientation ~v = + − + − (row one) and a 2-tangle with orientation ~v = + + −− (rows two and three). n -tangle that has been oriented so as to be compatible with T . Applying Theorem 2.3 toboth of these links immediately yields the following: Proposition 3.1.
Let φ be a local move that replaces the oriented n -tangle T with an n -tangle T whose endpoints are equivalently oriented. For any n -tangle T ′ , the polynomial f ( T ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by the gcd of the set { f ( e T B ( m )) − f ( e T B ( m )) } m ∈P n . Even for small n , Proposition 3.1 is of limited usage unless one may succinctly characterizethe closures e T B ( m ), e T B ( m ) for every m ∈ P n . In what follows, we consider various classes oflocal moves where this characterization is tractable, restricting from virtual links to classicallinks as needed. These classes will encompass, as special cases, oriented versions for manyof the local moves considered by Ganzell [2].Pause to observe that, if two links L , L are related via a finite sequence of φ -moves,repeated application of Proposition 3.1 says that f ( L ) − f ( L ) must be divisible by the great-est common divisor of the { f ( e T B ( m )) − f ( e T B ( m )) } m ∈P n . In cases where the e T B ( m ), e T B ( m )are easily computable for all m ∈ P n , this provides a necessary condition for determiningwhether φ represents an unlinking operation: Corollary 3.2.
Let φ be a local move that replaces the oriented n -tangle T with an n -tangle T whose endpoints are equivalently oriented, and let (cid:13) k be the unlink of k components. Thenthe virtual link L may be transformed into (cid:13) k via a finite sequence of φ -moves only if f ( L ) − f ( (cid:13) k ) = f ( L ) − ( − A − − A ) k − is divisible by the gcd of the { f ( e T B ( m )) − f ( e T B ( m )) } m ∈P n . .1 Rotational Local Moves of Classical n -tangles In this subsection we restrict our attention to local moves T T that involve rotation of T by some fixed angle. So let T be an n -tangle, and let r k ( T ) be the n -tangle that resultsfrom rotating T by kπn radians in the clockwise direction. For every T and every k >
0, thisdefines a local move T r k ( T ) that may or may not preserve orientation on endpoints.To ensure that our local moves preserve orientation, we restrict our attention to n -tangleswith standard orientation ~v = (+ − ) n and modify the rotational operator as follows. If k iseven, r k ( T ) already has standard orientation and we define ρ k ( T ) = r k ( T ). If k is odd, r k ( T )has the opposite orientation ~v ′ = ( − +) n . In this case we define ρ k ( T ) to be the n -tangle oforientation ~v that results from reversing every strand in r k ( T ), including closed strands thatdo not terminate at the boundary. Observe that f ( r k ( T )) = f ( ρ k ( T )) for any T , as reversingevery strand in a tangle fixes the writhe of every crossing.Now take m ∈ P n . For every k ≥ r k ( m ) to be the matching thatresults from a counterclockwise rotation of m by kπn radians. Clearly r k ( m ) is noncrossing forall k > m is noncrossing. For all m ∈ P n , notice that the closure T ( r k ( m ))may be obtained from T ( m ) via a counterclockwise rotation of T ( m ) − T by kπn radians.If T has standard orientation, we immediately have ρ k ( T )( m ) = T ( r k ( m )) for all m ∈ P n and all k >
0. However, for matchings with crossings this equality breaks down when wereplace the neighborhood of virtual crossings as in Figure 4. If we further assume that m is noncrossing, we always have e T B ( m ) = T B ( m ) = T ( m ) and may still assert ^ ρ k ( T ) B ( m ) = e T B ( r k ( m )) for every k >
0. All of this gives the following specialization of Proposition 3.1.
Proposition 3.3.
Let T be a classical n -tangle with standard orientation. For any classical n -tangle T ′ and every k > , the polynomial f ( ρ k ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by the gcdof the set { f ( T ( r k ( m ))) − f ( T ( m )) } m ∈M n . Looking to apply Proposition 3.3, we first consider rotational local moves T ρ k ( T )where T is a classical 2-tangle. Here we denote the two non-crossing matchings on 4 pointsby m = ((1 , , (3 , m = ((1 , , (2 , T is a classical 2-tangle with standardorientation, the local move T ρ ( T ) corresponds to the traditional notation of mutation.This means that following proposition is standard orientation version of the classic resultstating that the auxiliary polynomial is invariant under mutation. Proposition 3.4.
Let T be a classical -tangle with standard orientation. For any -tangle T ′ with compatible orientation, f ( ρ ( T ) ∪ T ′ ) = f ( T ∪ T ′ ) .Proof. Observe that r ( m ) = m and r ( m ) = m , giving f ( T ( r ( m ))) − f ( T ( m )) = 0and f ( T ( r ( m ))) − f ( T ( m )) = 0. Proposition 3.3 then implies that f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ )is divisible by 0 for any T ′ .Our next local move represents a “semi-mutation” on the associated diagram. For ademonstration of how the closures of a 2-tangle behave under this move, see Figure 6. Theorem 3.5.
Let T be a classical -tangle with standard orientation. For any -tangle T ′ with compatible orientation, f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by f ( T ( m )) − f ( T ( m )) . roof. Here we have r ( m ) = m and r ( m ) = m . Thus f ( T ( r ( m ))) − f ( T ( m )) = f ( T ( m )) − f ( T ( m )) and f ( T ( r ( m ))) − f ( T ( m )) = f ( T ( m )) − f ( T ( m )), from whichProposition 3.3 gives the desired result. T
12 3 4 ⇔ T r
12 3 4 = T r
23 4 1 T
12 3 4 ⇔ T r
12 3 4 = T r
23 4 1
Figure 6: Both closures T ( m i ) ρ ( T )( m i ) for a 2-tangle T undergoing the local move ofTheorem 3.5. Here T r denotes that the orientation of every strand in T has been reversed. Example 3.6.
Let T contain an even number k of positive half-twists, giving our local move T ρ ( T ) of the form ⇔ Here T ( m ) is the unknot, while T ( m ) is the (2 , k ) -torus link L (2 ,k ) . Theorem 3.5 thenstates f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by − f ( L (2 ,k ) ) for any -tangle T ′ , where f ( L (2 ,k ) ) = ( − k +1 A − k − (1 − A − + ( − k ( A − − k − A − − k ))1 − A − . By construction, this di-visor − f ( L (2 ,k ) ) is maximal for all classical links with respect to T ρ ( T ) . Example 3.7.
Let T contain any number k of positive half-twists followed by a clasp, giving T ρ ( T ) of the form ⇔ Here T ( m ) = T k is the twist knot with k half-twists, and T ( m ) = H is the Hopf link.It follows that f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by f ( T k ) − f ( H ) for any T ′ , where f ( H ) = − A − − A and ( T k ) = A − + ( − k A − k − ( − k +1 A − k − − A + 1 k odd − A − A − ( − k A − k + ( − k A − k − A + 1 k evenBy construction, this divisor f ( T k ) − f ( H ) is maximal for all classical links with respect tothis particular local move T ρ ( T ) . We now expand our attention to rotational local moves on classical n -tangles for n > M toderive a specialization of Proposition 3.3 whose divisibility conditions are especially simple.Denote the five elements of M by m a = ((1 , , (3 , , (5 , m a = ((1 , , (2 , , (4 , m b = ((1 , , (3 , , (4 , m b = ((1 , , (2 , , (5 , m b = ((1 , , (2 , , (3 , T ρ ( T ), a π -radian rotation whose effect on these variousclosures is shown in Figure 7. Theorem 3.8.
Let T be a classical -tangle with standard orientation. For any -tangle T ′ with compatible orientation, f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by f ( T ( m a )) − f ( T ( m a )) .Proof. We have r ( m a ) = m a , r ( m a ) = m a , and r ( m b i ) = m b i for i = 1 , ,
3. It followsthat f ( T ( r ( m a ))) − f ( T ( m a )) = f ( T ( m a )) − f ( T ( m a )) = − f ( T ( r ( m a ))) + f ( T ( m a ))and f ( T ( r ( m b i ))) − f ( T ( m b i )) = 0 for i = 1 , ,
3. The theorem then follows from Proposition3.3. T
123 4 5 6 T ( m a ) → ρ ← T
123 4 5 6 T ( m a ) T
123 4 5 6 T ( m b ) T
123 4 5 6 T ( m b ) T
123 4 5 6 T ( m b ) x ρ x ρ x ρ Figure 7: Up to reversal of all strands, the way in which the local move T ( m i ) ρ ( T )( m i )from Theorem 3.8 permutes the five closures of the classical 3-tangle T .Perhaps the best known example of a local move involving 3-tangles is the ∆-move, anunknotting operation whose unorientated form is shown in Figure 8. As shown by Murakamiand Nakanishi [10], the ∆-move possesses the interesting property that two links L , L maybe connected by a single ∆-move (of any orientation) if and only if L , L may be connectedby a single oriented ∆-move with standard orientation. Since a ∆-move with standardorientation qualifies as a local move of the form T ρ ( T ), Theorem 3.8 may then be12pplied to give a significantly simpler proof of the ∆-move divisibility theorem presented byGanzell [2]: Theorem 3.9.
Let L and L ′ be a pair of classical links that are separated by a single ∆ -move. Then f ( L ) − f ( L ′ ) is divisible by A − A − A + 1 . Furthermore, this divisor ismaximal for all classical links with respect to the ∆ -move.Proof. For a standard orientation version of the tangle on the left side of Figure 8, T ( m a )is the unknot and T ( m a ) is the left-handed trefoil. In the case of standard orientation,Theorem 3.8 then states that f ( L ) − f ( L ′ ) is then divisible by f (1) − f (3 ) = A − A − A +1.The general result follows from the observation of Murakami and Nakanishi [10]. The factthat this divisor is maximal follows from the fact that it equals f (1) − f (3 ). ⇔ Figure 8: The ∆ move.The method of Theorem 3.8 may be adapted to local moves T ρ ( T ) where the classical3-tangle is rotated by π -radians. However, due to the manner in which a π -radian rotationpermutes the various closures of T , the result is somewhat less elegant: Theorem 3.10.
Let T be a classical -tangle with standard orientation. For any -tangle T ′ with compatible orientation, f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisible by the greatest commondivisor of f ( T ( m b )) − f ( T ( m b )) and f ( T ( m b )) − f ( T ( m b )) .Proof. For this rotation r ( m a ) = m a , r ( m a ) = m a , r ( m b ) = b , r ( m b ) = m b , and r ( m b ) = m b . By Proposition 3.3, it follows that f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) is divisibleby the greatest common divisor of f ( T ( m b )) − f ( T ( m b )), f ( T ( m b )) − f ( T ( m b )), and f ( T ( m b )) − f ( T ( m b )). The result follows from the fact that any divisor of those first twodifferences is necessarily a divisor of the third difference. ∆ -move For the remainder of this paper, we explore a handful of additional local moves to which theresults of Subsection 3.1 do not immediately apply.We begin with the double-∆-move for classical 6-tangles, whose unoriented form is shownin Figure 9. As originally defined by Naik and Stanford [11], the double-∆-move is assumedto involve tangles where each pair of parallel strands are oriented in opposite directions.It is straightforward to show that two links L , L may be connected by a finite sequenceof double-∆-moves if and only if L , L may be connected by a finite sequences of double-∆-moves that all involve standard orientation 6-tangles. As shown in Figure 10, any non-standard orientation double-∆-move may be bypassed by performing a Reidemeister II moveon any pair of strands that aren’t in the proper orientation, and then redefining the tangle13 ⇔ Figure 9: The double-∆-move. +-+--+ + - + - -+ ⇒ f ~v + - + - -++-+--+ ⇓ R ⇑ R +-+--+ + - + - -+ ⇒ f + - + - -++-+--+ Figure 10: Replacing a non-standard orientation double-∆-move f ~v with a standard-orientation double-∆-move f in a manner that does not change the underlying links.boundary to obtain a standard orientation 6-tangle. As was the case with the original ∆-move, this technique allows us to prove a general divisibility result for the double-∆-movemerely by checking divisibility of f ( L ) − f ( L ) in the case of standard standard orientation. Theorem 3.11.
Let L and L ′ be a pair of classical links that are separated by a singledouble- ∆ -move. Then f ( L ) − f ( L ′ ) is divisible by A − A + A − A − A + A − A + 1 .Furthermore, this divisor is maximal for all classical links with respect to the double- ∆ -move.Proof. As the double-∆-move involves classical tangles with standard orientation, Proposi-tion 3.1 requires that we determine f ( T ( m )) − f ( T ( m )) for all C = 132 elements of M .However, using the endpoint numbering shown in Figure 9 , it is clear that T ( m ) = T ( m )for any m ∈ M that includes an arc of the form ( i, i + 1) for at least one odd integer i .This leaves fifteen closures for which f ( T ( m )) − f ( T ( m )) may be nonzero.Now observe that both tangles T , T involved in the double-∆-move are invariant underrotation by four strands. It follows that f ( T ( m )) − f ( T ( m )) = f ( T ( m ′ )) − f ( T ( m ′ )) forany pair of closures m, m ′ ∈ M that differ via rotation by four strands. Among our fifteenremaining closures, this reduces the necessary computations to the seven closures below.14 Among these seven remaining closures, it may be shown that we still have T ( m ) = T ( m )for all three closures in the top row. For the first two closures in the second row, f ( T ( m ))and f ( T ( m )) are A + A + A + 1 and A + A + A + 2 − A − + A − − A − (insome order). For the last two closures in the second row, T ( m ) and T ( m ) are a 2-cableof the unknot and a 2-cable of the trefoil (in some order), giving f ( T ( m )) and f ( T ( m )) of − A − A + A − A + A − A and − A − A − A + A − + A − − A − . Taking thegreatest common divisor of these non-trivial differences gives the desired result.To see that our divisor is maximal among all classical links, notice that A − A + A − A − A + A − A + 1 = f (1) − f ( K ), where K = 11 is the Kinoshita-Terasaka knot. Asthe Kinoshita-Terasaka has the same Alexander polynomial as the unknot, it follows fromthe work of Naik and Stanford [11] that it may be transformed into the unknot via a finitesequence of double-∆-moves.One significant application of Theorem 3.11 involves S-equivalence of knots. A pair ofclassical knots K, K ′ are said to be S-equivalent if their Seifert matrices are related bya sequence of elementary enlargements and similarity. Knots in the same S-equivalenceclass share many interesting properties, such as having identical Alexander polynomials andisometric Blanchfield pairings. More pertinent to this paper is the work of Naik and Stanford[11], who showed that two oriented classical knots are S-equivalent if and only if they arerelated by a sequence of double-∆-moves. This fact immediately prompts the followingcorollary of Theorem 3.11: Corollary 3.12.
Let
K, K ′ be classical knots such that K and K ′ are S-equivalent. Then f ( K ) − f ( K ′ ) is divisible by A − A + A − A − A + A − A + 1 . Most of the results in this section do not easily generalize to virtual tangles. This derivesfrom the fact that the virtual move of Figure 4, which was necessary for our derivation ofTheorem 2.3, gives all virtual crossings in T ( m ) − T a “preferred” quadrant that is fixed as T undergoes the local move. In particular, for the rotational move T ρ k ( T ), the modifiedclosures of Theorem 2.3 do not obey ^ ρ k ( T ) B ( m ) = e T B ( r k ( m )) if m contains at least onevirtual crossing. 15uckily, some local moves T T involving virtual tangles are simple enough that it ispossible to ignore Theorem 2.3 and manually calculate a similar, better-suited decompositionfor f ( T ∪ T ′ ) and f ( T ∪ T ′ ). One such move is a generalization of the “semi-mutation”move from Theorem 3.5 to virtual 2-tangles. Theorem 3.13.
Let T be a virtual -tangle with standard orientation. For any -tangle T ′ with compatible orientation, f ( T ∪ T ′ ) − f ( ρ ( T ) ∪ T ′ ) is divisible by f ( T ( m )) − f ( T ( m )) .Proof. Denoting the closures as in Figure 2, Proposition 2.1 immediately gives the followingdecompositions, where q i ∈ Z [ A, A − ]: h T ∪ T ′ i = q h T ( m ) i + q h T ( m ) i + q h T ( m ) ih ρ ( T ) ∪ T ′ i = q h ρ ( T ( m )) i + q h ρ ( T ( m )) i + q h ρ ( T ( m )) i = q h T ( m ) i + q h T ( m ) i + q h T ( m )) i Translating from the Kauffman bracket to the auxiliary polynomial requires that wereplace T ( m ) with closures that respect the standard orientation. The Kauffman skeinrelation gives h T ( m ) i = A − h T ( m v ) i− A − h T ( m ) i , with T ( m v ) as shown in the upper-rightcorner of Figure 5. Absorbing the various writhe terms ( − A ) − w from each f ( T ( m i )) into thethe leading Laurent polynomials gives the following decompositions, where p i ∈ Z [ A, A − ]and T r denotes the tangle obtained by reversing every strand in T . f ( T ∪ T ′ ) = p f ( T ( m )) + p f ( T ( m )) + [ p f ( T ( m v )) − p f ( T ( m ))] f ( ρ ( T ) ∪ T ′ ) = p f ( T r ( m )) + p f ( T r ( m )) + [ p f ( T r ( m v )) + p f ( T r ( m ))]Noting that f ( T r ( m )) = f ( T ( m )) for any closure m , we conclude that f ( ρ ( T ) ∪ T ′ ) − f ( T ∪ T ′ ) must be divisible by f ( T ( m )) − f ( T ( m )). References [1] A. Fish and E. Keyman, Jones polynomial invariants,
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