On a certain numerical invariant of link types

Murasugi, Kunio

doi:10.1090/s0002-9947-1965-0171275-5

Cited by 299 publications

(225 citation statements)

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“…(1) If α is the trivial representation, then Theorem 1.1 was proved in the one-variable case by Murasugi [Mu67] (cf. also [Ka77]) and in the multi-variable case it was proved independently by Kawauchi We prove a generalization of Milnor's classical duality theorem for Reidemeister torsion [Mi62] and use it as a key ingredient in the proof of the above sliceness obstruction.…”

Section: Remarkmentioning

confidence: 99%

Twisted torsion invariants and link concordance

Choon

Friedl

2011

Forum Mathematicum

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Abstract. The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor to the non-acyclic case. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.

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Section: Remarkmentioning

confidence: 99%

Twisted torsion invariants and link concordance

Choon

Friedl

2011

Forum Mathematicum

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“…Then the main theorem of this paper is of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700026094 [9] On the Alexander polynomial 325…”

Section: Statement Of Main Theoremmentioning

confidence: 99%

On the Alexander polynomial of alternating algebraic knots

Murasugi

1985

J Aust Math Soc A

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“…After collecting some knot and number theoretic preliminaries in Sections 2.1 and 2.2 resp., we begin in Section 3 with recalling a criterion for the Alexander polynomial of an achiral knot via the determinant, which follows from Murasugi's work on the signature [41] and the Lickorish-Millett [36] value of the Jones polynomial. These conditions show that, paradoxically formulated, although the Alexander polynomial cannot distinguish between a knot and its mirror image, it can still sometimes show that they are distinct.…”

Section: Introductionmentioning

confidence: 99%

Square numbers, spanning trees and invariants of achiral knots

Stoimenow¹

2005

Communications in Analysis and Geometry

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We give constructions to realize an odd number, which can be represented as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later, we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete result for perfect squares. Using the checkerboard coloring, then an application is given to the number of spanning trees in planar self-dual graphs. Another application is some enumeration results on achiral rational knots.

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On a certain numerical invariant of link types

Cited by 299 publications

References 12 publications

Twisted torsion invariants and link concordance

Twisted torsion invariants and link concordance

On the Alexander polynomial of alternating algebraic knots

Square numbers, spanning trees and invariants of achiral knots

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