The SL(2, <i>C</i>) Casson Invariant for Knots and the <i>Â</i>-polynomial

Bodén, Hans; Curtis, Cynthia L.

doi:10.4153/cjm-2015-025-5

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…As we noted in Section 5.1, has been conjectured to hold for all knots in

S^{3}

. This has been verified for many examples, including all two‐bridge knots and torus knots; see [, Section 2]. It has also been verified by Le and Tran for the

(- 2, 3, 2 n + 1)

pretzel knot for all

n \in Z

.…”

Section: Some Computationsmentioning

confidence: 58%

“…The A-polynomial was introduced in [13] and has been extensively studied since then. The A-polynomial is a close relative; it was defined by Boyer-Zhang [7] and it also appears in the work of Boden and Curtis [4] from which we draw our exposition.…”

Section: Relation To the A-polynomialmentioning

confidence: 97%

“…From a practical standpoint, Corollary does not help with computing

{italicHP}_{τ}^{*} false(K false)

unless one is able to get a handle on the

\overset{A}{true}

‐polynomial of

K

. Unfortunately, there appears to be little in the way of general computations of the

\overset{A}{true}

‐polynomial in the literature (but see for the

\overset{A}{true}

‐polynomial of certain Whitehead doubles). In contrast, the ordinary

A

‐polynomial is known to be effectively computable, and has been computed for wide classes of examples.…”

Section: Some Computationsmentioning

confidence: 99%

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A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three‐manifolds, similar to the three‐manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf‐theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two‐bridge and torus knots, the SL(2,C) invariant is determined by the l‐degree of the trueÂ‐polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.

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“…As we noted in Section 5.1, has been conjectured to hold for all knots in

S^{3}

. This has been verified for many examples, including all two‐bridge knots and torus knots; see [, Section 2]. It has also been verified by Le and Tran for the

(- 2, 3, 2 n + 1)

pretzel knot for all

n \in Z

.…”

Section: Some Computationsmentioning

confidence: 58%

Section: Relation To the A-polynomialmentioning

confidence: 97%

“…From a practical standpoint, Corollary does not help with computing

{italicHP}_{τ}^{*} false(K false)

unless one is able to get a handle on the

\overset{A}{true}

‐polynomial of

K

. Unfortunately, there appears to be little in the way of general computations of the

\overset{A}{true}

‐polynomial in the literature (but see for the

\overset{A}{true}

‐polynomial of certain Whitehead doubles). In contrast, the ordinary

A

‐polynomial is known to be effectively computable, and has been computed for wide classes of examples.…”

Section: Some Computationsmentioning

confidence: 99%

See 1 more Smart Citation

A sheaf‐theoretic SL(2,C) Floer homology for knots

Côté

Manolescu

2019

Proc. London Math. Soc.

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On SL(3, $${\varvec{\mathbb {C}}}$$ C )-representations of the Whitehead link group

Guilloux

Will

2018

Geom Dedicata

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We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,C)-character variety of π.

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On the sheaf-theoretic $\operatorname{SL}(2,\mathbb{C})$ Casson–Lin invariant

Côté

Neithalath

2022

J. Math. Soc. Japan

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The SL(2, C) Casson Invariant for Knots and the Â-polynomial

Cited by 3 publications

References 19 publications

A sheaf‐theoretic SL(2,C) Floer homology for knots

A sheaf‐theoretic SL(2,C) Floer homology for knots

On SL(3, $${\varvec{\mathbb {C}}}$$ C )-representations of the Whitehead link group

On the sheaf-theoretic $\operatorname{SL}(2,\mathbb{C})$ Casson–Lin invariant

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