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Erschienen in:
A Survey of Hyperbolic Knot Theory Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

Extending the Classical Skein

verfasst von : Louis H. Kauffman, Sofia Lambropoulou

Erschienen in: Knots, Low-Dimensional Topology and Applications

Verlag: 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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Metadaten
Titel
Extending the Classical Skein
verfasst von
Louis H. Kauffman
Sofia Lambropoulou
Copyright-Jahr
2019
Buch
Knots, Low-Dimensional Topology and Applications

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

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

Copyright-Jahr: 2019

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