KNOTS IN THE SOLID TORUS UP TO 6 CROSSINGS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250106 ◽  
Author(s):  
BOŠTJAN GABROVŠEK ◽  
MACIEJ MROCZKOWSKI
Keyword(s):  
Jones Polynomial ◽  
Solid Torus ◽  
Skein Modules ◽  
Skein Module ◽  
Almost All

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

THE (2, ∞)-SKEIN MODULE OF WHITEHEAD MANIFOLDS

1995 ◽  
Vol 04 (03) ◽  
pp. 411-427 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Skein Modules ◽  
Skein Module ◽  
Natural Filtration ◽  
Stark Contrast ◽  
Torsion Free

To any orientable 3-manifold one can associate a module, called the (2, ∞)-skein module, which is essentially a generalization of the Jones polynomial of links in S3. For an uncountable collection of open contractible 3-manifolds, each constructed in a fashion similar to the classic Whitehead manifold, we prove that their (2, ∞)-skein modules are infinitely generated, torsion free, but not free. These examples stand in stark contrast to [Formula: see text], whose (2, ∞)-skein module is free on one generator. To each of these manifolds we associate a subgroup G of the rationals which may be interpreted via wrapping numbers. We show that the skein module of M has a natural filtration by modules indexed by G. For the specific case of the Whitehead manifold, we describe its (2, ∞)-skein module and associated filtration in greater detail.

scholarly journals Topological steps toward the Homflypt skein module of the lens spaces L(p,1) via braids

2016 ◽  
Vol 25 (14) ◽  
pp. 1650084 ◽  
Author(s):  
Ioannis Diamantis ◽  
Sofia Lambropoulou ◽  
Jozef H. Przytycki
Keyword(s):  
Infinite System ◽  
Solid Torus ◽  
Lens Spaces ◽  
System Of Equations ◽  
Closed Curves ◽  
Skein Modules ◽  
Skein Module ◽  
Basic Set ◽  
Knots And Links ◽  
Shed Light

In this paper, we work toward the Homflypt skein module of the lens spaces [Formula: see text], [Formula: see text] using braids. In particular, we establish the connection between [Formula: see text], the Homflypt skein module of the solid torus ST, and [Formula: see text] and arrive at an infinite system, whose solution corresponds to the computation of [Formula: see text]. We start from the Lambropoulou invariant [Formula: see text] for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, [Formula: see text], of [Formula: see text] presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra 220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014 , arXiv:1412.3642 [math.GT]]. We show that [Formula: see text] is obtained from [Formula: see text] by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis [Formula: see text], where the bbm are performed on any moving strand of each element in [Formula: see text]. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set [Formula: see text]. The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. [Formula: see text]-manifolds, since any [Formula: see text]-manifold can be obtained by surgery on [Formula: see text] along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.

scholarly journals ON THE S1 × S2 HOMFLY-PT INVARIANT AND LEGENDRIAN LINKS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350040 ◽  
Author(s):  
MIKHAIL LAVROV ◽  
DAN RUTHERFORD
Keyword(s):  
Rational Function ◽  
Contact Structure ◽  
Solid Torus ◽  
Laurent Polynomial ◽  
Skein Modules ◽  
Skein Module ◽  
Tight Contact ◽  
Legendrian Links ◽  
Simple Basis

In [On the HOMFLY-PT skein module of S1 × S2, Math. Z. 237(4) (2001) 769–814], Gilmer and Zhong established the existence of an invariant for links in S1 × S2 which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in S1 × S2 and shows that the invariant is in fact a Laurent polynomial in a and z = s – s-1. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston–Bennequin number to Legendrian links in S1 × S2 with its tight contact structure.

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

THE (2, ∞)-SKEIN MODULE OF LENS SPACES: A GENERALIZATION OF THE JONES POLYNOMIAL

1993 ◽  
Vol 02 (03) ◽  
pp. 321-333 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Lens Spaces ◽  
Skein Module

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.

scholarly journals Non-commutative trigonometry and the A-polynomial of the trefoil knot

2002 ◽  
Vol 133 (2) ◽  
pp. 311-323 ◽  
Author(s):  
RĂZVAN GELCA
Keyword(s):  
Character Variety ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Trefoil Knot ◽  
Sum Formula ◽  
Left And Right ◽  
Trefoil Knots ◽  
The Relationship ◽  
The Ideal

The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character variety of the fundamental group, which was explained in [2]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.

scholarly journals An important step for the computation of the HOMFLYPT skein module of the lens spaces L(p,1) via braids

2019 ◽  
Vol 28 (11) ◽  
pp. 1940007 ◽  
Author(s):  
Ioannis Diamantis ◽  
Sofia Lambropoulou
Keyword(s):  
Infinite System ◽  
Solid Torus ◽  
Lens Spaces ◽  
System Of Equations ◽  
Skein Module ◽  
Knots And Links

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces [Formula: see text] from the HOMFLYPT skein module of the solid torus, [Formula: see text], it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant [Formula: see text] for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set [Formula: see text], augmenting the basis [Formula: see text] of [Formula: see text].

A geometric proof that alternating knots are non-trivial

1991 ◽  
Vol 109 (3) ◽  
pp. 425-431 ◽  
Author(s):  
William Menasco ◽  
Morwen Thistlethwaite
Keyword(s):  
Jones Polynomial ◽  
Related Result ◽  
Slight Variation ◽  
Geometric Proof ◽  
Link Diagram ◽  
Combinatorial Argument ◽  
Algebraic Invariant ◽  
Almost All ◽  
Alternating Links ◽  
Alternating Link

There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

scholarly journals A new basis for the Homflypt skein module of the solid torus

2016 ◽  
Vol 220 (2) ◽  
pp. 577-605 ◽  
Author(s):  
Ioannis Diamantis ◽  
Sofia Lambropoulou
Keyword(s):  
Solid Torus ◽  
Skein Module

On lassos and the Jones polynomial of satellite knots

2016 ◽  
Vol 25 (02) ◽  
pp. 1650011
Author(s):  
Adrián Jiménez Pascual
Keyword(s):  
Jones Polynomial ◽  
Solid Torus ◽  
Alexander Polynomial ◽  
New Family ◽  
Jones Polynomials ◽  
Polynomial Formula ◽  
Satellite Knots

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot [Formula: see text] with Alexander polynomial [Formula: see text], I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial [Formula: see text] where [Formula: see text]. In particular, I prove that if [Formula: see text] these satellite knots have different Jones polynomials.

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