On the 𝑧-degree of the Kauffman polynomial of a tangle decomposition

FROM RIBBON CATEGORIES TO GENERALIZED YANG–BAXTER OPERATORS AND LINK INVARIANTS (AFTER KITAEV AND WANG)

International Journal of Mathematics ◽

10.1142/s0129167x12501261 ◽

2013 ◽

Vol 24 (01) ◽

pp. 1250126 ◽

Cited By ~ 1

Author(s):

SEUNG-MOON HONG

Keyword(s):

The Other ◽

Polynomial Invariant ◽

Link Invariants ◽

Fusion Categories ◽

New Family ◽

Kauffman Polynomial ◽

Twist Structure

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

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Kontsevich’s integral for the Kauffman polynomial

Nagoya Mathematical Journal ◽

10.1017/s0027763000005638 ◽

1996 ◽

Vol 142 ◽

pp. 39-65 ◽

Cited By ~ 40

Author(s):

Thang Tu Quoc Le ◽

Jun Murakami

Keyword(s):

Finite Type ◽

Knot Invariants ◽

Chord Diagrams ◽

Knot Invariant ◽

Kauffman Polynomial ◽

Zamolodchikov Equation

Kontsevich’s integral is a knot invariant which contains in itself all knot invariants of finite type, or Vassiliev’s invariants. The value of this integral lies in an algebra A0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11].

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