Nontrivial Alexander polynomials of knots and links

Friedl, Stefan; Vidussi, Stefano

doi:10.1112/blms/bdm048

Cited by 12 publications

(14 citation statements)

References 25 publications

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“…We thus see that twisted Alexander polynomials detect the unknot, and in a similar vein we showed in [FV07] twisted Alexander polynomials detect the Hopf link. It is natural to ask whether twisted Alexander modules characterize other classes of knots and links.…”

Section: Introduction and Main Resultssupporting

confidence: 83%

Twisted Alexander Invariants Detect Trivial Links

Friedl¹,

Vidussi²

2017

Can. math. bull.

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Section: Introduction and Main Resultssupporting

confidence: 83%

Twisted Alexander Invariants Detect Trivial Links

Friedl¹,

Vidussi²

2017

Can. math. bull.

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“…The early development of this invariant as a tool in classical knot theory, in which case X was taken to be a classical knot complement, appeared in such papers as [18,24,27,41]. The theory and application of twisted knot polynomials has been considered by many authors; a few papers include [3,10,12,14,15,17,25,26,37].…”

Section: Twisted Polynomialsmentioning

confidence: 99%

Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

2009

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Given a knot complement X and its p-fold cyclic cover X p → X , we identify twisted polynomials associated to G L 1 F[t ±1 ] representations of π 1 (X p ) with twisted polynomials associated to related G L p F[t ±1 ] representations of π 1 (X ) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P (3,7,9,11,15), corresponding to permutations of (7, 9, 11, 15), represent distinct concordance classes.

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“…In particular, as the 0-surgery of S 3 along K is fibered if and only if K is, we deduce the following (cf. also [FV06b]). …”

Section: Corollarymentioning

confidence: 99%

Symplectic $\mathbf{S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm

Friedl¹,

Vidussi²

2007

J. Amer. Math. Soc.

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Let N be a closed, oriented 3-manifold. A folklore conjecture states that S 1 × N admits a symplectic structure if and only if N admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes 3-manifolds with vanishing Thurston norm, graph manifolds and 3-manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic 3-manifolds). Our result covers, in particular, the case of 0framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the 3-manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.

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Nontrivial Alexander polynomials of knots and links

Cited by 12 publications

References 25 publications

Twisted Alexander Invariants Detect Trivial Links

Twisted Alexander Invariants Detect Trivial Links

Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Symplectic $\mathbf{S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm

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