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

    Fingerprint

    Knot
    Signature
    Diagram
    Trefoil
    Trivial
    Left handed
    Unknotting number
    Twist
    Exception
    Partial
    Generalise
    Zero

    Keywords

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

    ASJC Scopus subject areas

    • Algebra and Number Theory

    Cite this

    Almost positive links have negative signature. / Przytycki, Józef H.; Taniyama, Kouki.

    In: Journal of Knot Theory and its Ramifications, Vol. 19, No. 2, 02.2010, p. 187-289.

    Research output: Contribution to journalArticle

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