The Colored Jones Polynomial of Singular Knots

The study of singular knots, or equivalently rigid 4-valent graphs, and their invariants was generated largely by the theory of Vassiliev invariants. Many existing knot invariants have been extended to singular knot invariants. In [3], Birman introduced braids in the theory of Vassiliev via the singular braids and conjectured that the monoid of singular braids maps injectively into the group algebra of the braid group. A proof of this conjecture was given by Paris in [25]. Fiedler extended the Kauffman state models of the Jones and Alexander polynomials to the context of singular knots [6]. In [7] Gemein investigated extensions of the Artin representation and the Burau representation to the singular braid monoid and the relations between them. Juyumaya and Lambropoulou constructed a Jones-type invariant for singular links using a Markov trace on a variation of the Hecke algebra [15]. In [20] Kauffman and Vogel defined a polynomial invariant of embedded 4-valent graph in R3 extending an invariant for links in R3 called the Kauffman polynomial [17]. The latter is a two variable polynomial that takes value in Z[a, a−1, z] and is an invariant of regular isotopy for links. The Kauffman polynomial of a link L is denoted by [L] and is defined via the following axioms :

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