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New skein invariants of links

Department of Mathematics, Statistics and Computer Science, 851 South Morgan Street, University of Illinois at Chicago, Illinois 60607-7045, Chicago

Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia

E-mail Address: kauffman@uic.edu

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and

Sofia Lambropoulou

Department of Mathematics, National Technical University of Athens, Zografou Campus, GR-157 80 Athens, Greece

E-mail Address: sofia@math.ntua.gr

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https://doi.org/10.1142/S0218216519400182Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue: Knots, Low-Dimensional Topology and Applications — Part II

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.

Keywords:

AMSC: 57M27, 57M25

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