Skein Invariants of Links and Their State Sum Models
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 obtain...
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MDPI AG
2017-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/9/10/226 |
| _version_ | 1811279052013568000 |
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| author | Louis H. Kauffman Sofia Lambropoulou |
| author_facet | Louis H. Kauffman Sofia Lambropoulou |
| author_sort | Louis H. Kauffman |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T00:47:53Z |
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| id | doaj.art-7642386a247c4855a1008e0d82f66c11 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T00:47:53Z |
| publishDate | 2017-10-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-7642386a247c4855a1008e0d82f66c112022-12-22T03:09:57ZengMDPI AGSymmetry2073-89942017-10-0191022610.3390/sym9100226sym9100226Skein Invariants of Links and Their State Sum ModelsLouis H. Kauffman0Sofia Lambropoulou1Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USASchool of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, GreeceWe 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.https://www.mdpi.com/2073-8994/9/10/226classical linksmixed crossingsskein relationsstacks of knotsHomflypt polynomialKauffman polynomialDubrovnik polynomial3-variable skein link invariantclosed combinatorial formulastate sumsdouble state summationskein template algorithm |
| spellingShingle | Louis H. Kauffman Sofia Lambropoulou Skein Invariants of Links and Their State Sum Models Symmetry 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 |
| title | Skein Invariants of Links and Their State Sum Models |
| title_full | Skein Invariants of Links and Their State Sum Models |
| title_fullStr | Skein Invariants of Links and Their State Sum Models |
| title_full_unstemmed | Skein Invariants of Links and Their State Sum Models |
| title_short | Skein Invariants of Links and Their State Sum Models |
| title_sort | skein invariants of links and their state sum models |
| topic | 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 |
| url | https://www.mdpi.com/2073-8994/9/10/226 |
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