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Local moves on knots and products of knots

Volume 103 / 2014

Louis H. Kauffman, Eiji Ogasa Banach Center Publications 103 (2014), 159-209 MSC: 57N10, 57N13, 57N15. DOI: 10.4064/bc103-0-7

Abstract

We show a relation between products of knots, which are generalized from the theory of isolated singularities of complex hypersurfaces, and local moves on knots in all dimensions. We discuss the following problem. Let $K$ be a 1-knot which is obtained from another 1-knot $J$ by a single crossing change (resp. pass-move). For a given knot $A$, what kind of relation do the products of knots, $K\otimes A$ and $J\otimes A$, have? We characterize these kinds of relation between $K\otimes A$ and $J\otimes A$ by using local moves on high dimensional knots. We also discuss a connection between local moves and knot invariants. We show a new form of identities for knot polynomials associated with a local move.

Authors

  • Louis H. KauffmanDepartment of Mathematics, Statistics, and Computer Science
    University of Illinois at Chicago
    851 South Morgan Street
    Chicago, Illinois 60607-7045, USA
    e-mail
  • Eiji OgasaComputer Science
    Meijigakuin University
    Yokohama
    Kanagawa, 244-8539, Japan
    e-mail
    e-mail

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