Nine generators of the skein space of the
3–torus

Carrega, Alessio

doi:10.2140/agt.2017.17.3449

Cited by 9 publications

(11 citation statements)

References 19 publications

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“…We can choose (μ, ν) in equation (4) so that (x + μ, y + ν) = (α, β). As the vertex coefficients are nonzero, and as P is symmetric, equation 4…”

Section: A Stability Theorem Under Generic Dehn Filling Of Knotsmentioning

confidence: 99%

“…Because the coefficients c i α,β (A) are invertible whenever (α, β) is a vertex of P i , so are the vertex coefficients of the translated annihilating polynomials by equation (4).…”

Section: Corollary 52 Let L Be a Two Component Link Inmentioning

confidence: 99%

“…The skein module K(M, Z[A, A −1 ]) is often not finitely generated; in fact, by results of Bullock and Przytycki-Sikora [2,27], the evaluation A = −1 with ground ring C yields a surjective map to the coordinate ring of the SL 2 (C)-character variety of M . Nonetheless, a surprising conjecture of Witten states that the skein modules of closed 3-manifolds with Q(A)coefficients are finitely generated: The first written mention of this conjecture can be found in [4] and a detailed exposition in [11]. Many skein modules of closed 3-manifolds had already been computed (often with Z[A, A −1 ] coefficients) and in all those cases, the module K(M, Q(A)) is indeed finite dimensional.…”

Section: Introductionmentioning

confidence: 99%

“…The first written mention of this conjecture can be found in [4] and a detailed exposition in [11]. Many skein modules of closed 3‐manifolds had already been computed (often with

Z [A, A^{- 1}]

coefficients) and in all those cases, the module

K (M, Q (A))

is indeed finite dimensional.…”

Section: Introductionmentioning

confidence: 99%

“…Many skein modules of closed 3‐manifolds had already been computed (often with

Z [A, A^{- 1}]

coefficients) and in all those cases, the module

K (M, Q (A))

is indeed finite dimensional. We give a list of closed 3‐manifolds that are known to satisfy the conjecture:

S^{3}

(where

K (S^{3}, Q false(A false)) = Q false(A false)

; this is equivalent to the existence and unicity of the Kauffman bracket),

S^{2} \times S^{1}

and lens spaces by Hoste–Przytycki [14, 15], integer Dehn filling on the trefoil knot by Bullock [1], the quaternionic manifold by Gilmer and Harris [10], some Dehn fillings of

(2, 2 b) -

torus links by Harris [12], a family of prime prism manifolds by Mroczkowski [23] and the 3‐torus by Carrega [4]. Moreover Przytycki showed that for a connected sum of 3‐manifolds,

K false(M_{1} # M_{2}, double-struckQ (A) false) = K false(M_{1}, double-struckQ (A) false) \otimes K false(M_{2}, double-struckQ (A) false)

, hence Conjecture 1.1 is stable under connected sums.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Infinite families of hyperbolic 3‐manifolds with finite‐dimensional skein modules

Detcherry

2020

J. London Math. Soc.

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The Kauffman bracket skein module K(M) of a 3-manifold M is the quotient of the Q(A)vector space spanned by isotopy classes of links in M by the Kauffman relations. A conjecture of Witten states that if M is closed, then K(M) is finite dimensional. We introduce a version of this conjecture for manifolds with boundary and prove a stability property for generic Dehn filling of knots. As a result, we provide the first hyperbolic examples of the conjecture, proving that almost all Dehn fillings of any two-bridge knot satisfy the conjecture. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

show abstract

“…We can choose (μ, ν) in equation (4) so that (x + μ, y + ν) = (α, β). As the vertex coefficients are nonzero, and as P is symmetric, equation 4…”

Section: A Stability Theorem Under Generic Dehn Filling Of Knotsmentioning

confidence: 99%

“…Because the coefficients c i α,β (A) are invertible whenever (α, β) is a vertex of P i , so are the vertex coefficients of the translated annihilating polynomials by equation (4).…”

Section: Corollary 52 Let L Be a Two Component Link Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The first written mention of this conjecture can be found in [4] and a detailed exposition in [11]. Many skein modules of closed 3‐manifolds had already been computed (often with

Z [A, A^{- 1}]

coefficients) and in all those cases, the module

K (M, Q (A))

is indeed finite dimensional.…”

Section: Introductionmentioning

confidence: 99%

“…Many skein modules of closed 3‐manifolds had already been computed (often with

Z [A, A^{- 1}]

coefficients) and in all those cases, the module

K (M, Q (A))

is indeed finite dimensional. We give a list of closed 3‐manifolds that are known to satisfy the conjecture:

S^{3}

(where

K (S^{3}, Q false(A false)) = Q false(A false)

; this is equivalent to the existence and unicity of the Kauffman bracket),

S^{2} \times S^{1}

and lens spaces by Hoste–Przytycki [14, 15], integer Dehn filling on the trefoil knot by Bullock [1], the quaternionic manifold by Gilmer and Harris [10], some Dehn fillings of

(2, 2 b) -

torus links by Harris [12], a family of prime prism manifolds by Mroczkowski [23] and the 3‐torus by Carrega [4]. Moreover Przytycki showed that for a connected sum of 3‐manifolds,