scholarly journals The Chebyshev–Frobenius homomorphism for stated skein modules of 3-manifolds

2022 ◽  
Author(s):  
Wade Bloomquist ◽  
Thang T. Q. Lê
Keyword(s):  
Skein Modules

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.

scholarly journals Link bordism skein modules

2004 ◽  
Vol 184 ◽  
pp. 113-134 ◽  
Author(s):  
Uwe Kaiser
Keyword(s):  
Skein Modules

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 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.

On framings of links in 3-manifolds

2020 ◽  
pp. 1-13
Author(s):  
Rhea Palak Bakshi ◽  
Dionne Ibarra ◽  
Gabriel Montoya-Vega ◽  
Józef H. Przytycki ◽  
Deborah Weeks
Keyword(s):  
Main Tool ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Light Bulb ◽  
Class Groups ◽  
Skein Modules ◽  
Ambient Isotopy

Abstract We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$ -manifold is when the manifold has a properly embedded non-separating $S^{2}$ . This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of $3$ -manifolds. We also relate our results to the theory of skein modules.

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.

A survey of skein modules of 3-manifolds

Knots 90 ◽  
2014 ◽  
Author(s):  
Jim Hoste ◽  
J. H. Przytycki
Keyword(s):  
Skein Modules
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