On the Alexander polynomial of alternating algebraic knots

Murasugi, Kunio

doi:10.1017/s1446788700026094

Cited by 27 publications

(13 citation statements)

References 7 publications

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“…Other restrictions on the Alexander polynomials of alternating knots have been conjectured by Fox, see [9] (see also [23], where these properties are verified for a large class of alternating knots). Specifically, Fox conjectures that for an alternating knot, the absolute values of the coefficients of the Alexander polynomial |a s | are non-increasing in s, for s ≥ 0.…”

Section: Applications To the Topology Of Alternating Knotsmentioning

confidence: 90%

Heegaard Floer homology and alternating knots

2003

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In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

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Section: Applications To the Topology Of Alternating Knotsmentioning

confidence: 90%

Heegaard Floer homology and alternating knots

2003

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“…The signature of the symmetrized Seifert matrix gives a knot invariant σ(K) satisfying a number of basic properties [23]: it is additive under connected sums, it changes in a controlled manner under crossing changes, and it gives a lower bound on the genus of a slice surface. Levine and Tristram [40] extend this invariant to a one-parameter family of knot invariants σ ω indexed by points ω on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

Concordance homomorphisms from knot Floer homology

Ozsváth

Stipsicz

Szabó

2017

Advances in Mathematics

151

231

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“…The case of two-bridge knots is confirmed by Hartley [2]. More generally Murasugi proved it for alternating algebraic knots [6]. The case of genus two alternating knots has also been verified by Ozsváth and Szabó using Heegaard Floer homology [7], and by Jong via a combinatorial method [4].…”

Section: Conjecture 1 ([1]mentioning

confidence: 81%

On the Alexander polynomial and the signature invariant of two-bridge knots

Chen¹

2017

Preprint

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Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.

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On the Alexander polynomial of alternating algebraic knots

Abstract: A conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is proved for alternating algebraic (or arborescent) knots, which include two-bridge knots.1980 Mathematics subject classification (Amer. Math. Soc): 57 M 25.

Cited by 27 publications

References 7 publications

Heegaard Floer homology and alternating knots

Heegaard Floer homology and alternating knots

Concordance homomorphisms from knot Floer homology

On the Alexander polynomial and the signature invariant of two-bridge knots

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