Mutation of knots

Kearton, C.

doi:10.1090/s0002-9939-1989-0929430-1

Cited by 10 publications

(6 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Kearton [20] used these examples to show that mutation acts nontrivially on concordance. These examples were built specifically so that the Casson-Gordon method could be applied.…”

Section: Pretzel Knotsmentioning

confidence: 99%

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

2009

View full text Add to dashboard Cite

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.

show abstract

“…Kearton [20] used these examples to show that mutation acts nontrivially on concordance. These examples were built specifically so that the Casson-Gordon method could be applied.…”

Section: Pretzel Knotsmentioning

confidence: 99%

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

2009

View full text Add to dashboard Cite

show abstract

“…A particularly interesting knot in this list is 11n 34 , which is topologically slice (since it has Alexander polynomial 1), but it is not known whether it is smoothly slice. In fact, it is a mutant of 11n 42 which is smoothly slice, but as shown by Kearton [5] mutation does not preserve concordance class, so 11n 34 is not necessarily smoothly slice.…”

Section: Summary Of Resultsmentioning

confidence: 98%

The Concordance Genus of 11-Crossing Knots

Kearney

2013

J. Knot Theory Ramifications

View full text Add to dashboard Cite

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11-crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

show abstract

“…The signature is also invariant under mutation [c.f. [Kea89]]. For pretzel knots, if we combine this fact with (1) we see that computation of the signature of K " P pp 1 , ..., p k q may be obtained using any knot in P tp 1 , ..., p k u.…”

Section: The Signature Condition and Proof Of Theoremmentioning

confidence: 97%

Slice implies mutant ribbon for odd 5–stranded pretzel knots

Bryant

2017

Algebr. Geom. Topol.

View full text Add to dashboard Cite

A pretzel knot K is called odd if all its twist parameters are odd, and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5stranded pretzel knots are mutant ribbon. We do this in stages by first showing that 5stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. Next, we show that any odd, 5stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.

show abstract

Mutation of knots

Abstract: In general, mutation does not preserve the Alexander module or the concordance class of a knot.

Cited by 10 publications

References 6 publications

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

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

The Concordance Genus of 11-Crossing Knots

Slice implies mutant ribbon for odd 5–stranded pretzel knots

Contact Info

Product

Resources

About