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Displaying similar documents to “On the AJ conjecture for cables of twist knots”

Nonfibered knots and representation shifts

Daniel S. Silver, Susan G. Williams (2009)

Banach Center Publications

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A conjecture of [swTAMS] states that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.

Positive knots, closed braids and the Jones polynomial

Alexander Stoimenow (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Using the recent Gauß diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no...

The writhes of a virtual knot

Shin Satoh, Kenta Taniguchi (2014)

Fundamenta Mathematicae

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Kauffman introduced a fundamental invariant of a virtual knot called the odd writhe. There are several generalizations of the odd writhe, such as the index polynomial and the odd writhe polynomial. In this paper, we define the n-writhe for each non-zero integer n, which unifies these invariants, and study various properties of the n-writhe.

Unknotting number and knot diagram.

Yasutaka Nakanishi (1996)

Revista Matemática de la Universidad Complutense de Madrid

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This note is a continuation of a former paper, where we have discussed the unknotting number of knots with respect to knot diagrams. We will show that for every minimum-crossing knot-diagram among all unknotting-number-one two-bridge knot there exist crossings whose exchange yields the trivial knot, if the third Tait conjecture is true.

Every knot is a billiard knot

P. V. Koseleff, D. Pecker (2014)

Banach Center Publications

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We show that every knot can be realized as a billiard trajectory in a convex prism. This proves a conjecture of Jones and Przytycki.

Minimal degree sequence for 2-bridge knots

Prabhakar Madeti, Rama Mishra (2006)

Fundamenta Mathematicae

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We discuss polynomial representations for 2-bridge knots and determine the minimal degree sequence for all such knots. We apply the connection between rational tangles and 2-bridge knots.

A twisted dimer model for knots

Moshe Cohen, Oliver T. Dasbach, Heather M. Russell (2014)

Fundamenta Mathematicae

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We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot depending on a representation of the knot group.

Virtual biquandles

Louis H. Kauffman, Vassily O. Manturov (2005)

Fundamenta Mathematicae

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We describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle [KR], [FJK], the virtual quandle [Ma2], the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew [BF], the concepts and properties of long virtual knots [Ma10], and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual...