New skein invariants of links

Louis H. Kauffman, Sofia Lambropoulou

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1 Citation (Scopus)

Abstract

We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T) on sublinks of a given link L, obtained by partitioning L into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.

Original languageEnglish
Article number1940018
Number of pages53
JournalJournal of Knot Theory and its Ramifications
Volume28
Issue number13
DOIs
Publication statusPublished - 1 Nov 2019

Keywords

  • 3-variable link invariant
  • Classical links
  • closed combinatorial formulae
  • Dubrovnik polynomial
  • Homflypt polynomial
  • Kauffman polynomial
  • mixed crossings
  • Reidemeister moves
  • skein invariants
  • skein relations
  • stacks of knots
  • Yokonuma-Hecke algebras
  • POLYNOMIAL INVARIANT
  • BRAIDS
  • HECKE ALGEBRAS
  • FRAMIZATION
  • ISOMORPHISM THEOREM
  • KNOTS
  • MODULE
  • REPRESENTATION-THEORY

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