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Published in:
A Survey of Hyperbolic Knot Theory Read chapter Read first chapter

2019 | OriginalPaper | Chapter

Extending the Classical Skein

Authors : Louis H. Kauffman, Sofia Lambropoulou

Published in: Knots, Low-Dimensional Topology and Applications

Publisher: Springer International Publishing

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Abstract

We summarize the theory of a new skein invariant of classical links H[H] that generalizes the regular isotopy version of the Homflypt polynomial, H. The invariant H[H] is based on a procedure where we apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots and then we evaluate the resulting knots using the invariant H and inserting at the same time a new parameter. This procedure, remarkably, leads to a generalization of H but also to generalizations of other known skein invariants, such as the Kauffman polynomial. We discuss the different approaches to the link invariant H[H], the algebraic one related to its ambient isotopy equivalent invariant \(\Theta \), the skein-theoretic one and its reformulation into a summation of the generating invariant H on sublinks of a given link. We finally give examples illustrating the behaviour of the invariant H[H] and we discuss further research directions and possible application areas.

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previous chapter A Survey of Grid Diagrams and a Proof of Alexander’s Theorem
next chapter From the Framisation of the Temperley–Lieb Algebra to the Jones Polynomial: An Algebraic Approach
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Metadata
Title
Extending the Classical Skein
Authors
Louis H. Kauffman
Sofia Lambropoulou
Copyright Year
2019
Book
Knots, Low-Dimensional Topology and Applications

Print ISBN: 978-3-030-16030-2

Electronic ISBN: 978-3-030-16031-9

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-16031-9_11

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