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Open AccessFeature PaperArticle

Skein Invariants of Links and Their State Sum Models

by 1,† and 2,*,†
1
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA
2
School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2017, 9(10), 226; https://doi.org/10.3390/sym9100226
Received: 19 September 2017 / Revised: 2 October 2017 / Accepted: 9 October 2017 / Published: 13 October 2017
(This article belongs to the Special Issue Knot Theory and Its Applications)
We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θ d related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ , which generalizes the Homflypt polynomial. H [ H ] is the regular isotopy counterpart of Θ . The invariants K [ K ] and D [ D ] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally. View Full-Text
Keywords: classical links; mixed crossings; skein relations; stacks of knots; Homflypt polynomial; Kauffman polynomial; Dubrovnik polynomial; 3-variable skein link invariant; closed combinatorial formula; state sums; double state summation; skein template algorithm classical links; mixed crossings; skein relations; stacks of knots; Homflypt polynomial; Kauffman polynomial; Dubrovnik polynomial; 3-variable skein link invariant; closed combinatorial formula; state sums; double state summation; skein template algorithm
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MDPI and ACS Style

Kauffman, L.H.; Lambropoulou, S. Skein Invariants of Links and Their State Sum Models. Symmetry 2017, 9, 226.

AMA Style

Kauffman LH, Lambropoulou S. Skein Invariants of Links and Their State Sum Models. Symmetry. 2017; 9(10):226.

Chicago/Turabian Style

Kauffman, Louis H.; Lambropoulou, Sofia. 2017. "Skein Invariants of Links and Their State Sum Models" Symmetry 9, no. 10: 226.

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