Twisted Alexander polynomials and surjectivity of a group homomorphism

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguishing knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore we prove that these invariants decide if a genus one knot is fibered. Finally we also show that these invariants distinguish all mutants with up to 12 crossings.

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“…, hence there exists a polynomial p(t) ∈ Z[t ±1 ] such that Δ γ K = p(t)Δ β N K (cf. also [14]). In particular Δ γ X(K) = 0.…”

Section: Proof Of Theorem 13mentioning

confidence: 87%

Nontrivial Alexander polynomials of knots and links

Friedl¹,

Vidussi

2007

Bulletin of the London Mathematical Society

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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 a similar way, the divisibility results of [12] proved for Wada's invariant give the following obstruction to the existence of epimorphisms preserving distinguished elements.…”

Section: Alexander-lin Polynomialsmentioning

confidence: 77%