Almost positive links have negative signature

Józef H. Przytycki, Kouki Taniyama

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

Original languageEnglish
Pages (from-to)187-289
Number of pages103
JournalJournal of Knot Theory and its Ramifications
Volume19
Issue number2
DOIs
Publication statusPublished - 2010 Feb 1

Keywords

  • Almost positive link
  • Jones polynomial
  • Positive link
  • Signature
  • TristramLevine signature
  • Twist knot
  • Unknotting number

ASJC Scopus subject areas

  • Algebra and Number Theory

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