Skein-theoretical derivation of some formulas of Habiro. (English) Zbl 1042.57005

The purpose of this paper is to give alternative proofs of some formulae, first established by K. Habiro, identifying certain elements in the Kauffman bracket skein module \(\mathcal B\) of a solid torus. These elements are used by Habiro, in a recent paper studying the colored Jones polynomial, to construct certain quantum invariants of homology 3-spheres which generalize the Reshetikhin-Turaev and Ohtsuki invariants. These elements \(\omega_+\) and \(\omega_-\) are characterized by the property that when they encircle an even number of strands of any link they have the effect of giving those strands a positive or negative twist (with respect to the Kauffman bracket). Habiro computes these elements and their powers by expressing them as an explicit linear combination of certain basis elements of \(\mathcal B\). His proof uses the quantum groups \(U_q \mathfrak{sl}_2\). In the present paper these formulae are derived using only skein-theoretic methods similar to the techniques in the series of papers of Blanchet, Habegger, Vogel and the author.
As an application some formulae for the colored Jones polynomial of twist knots are derived, generalizing computations of Habiro.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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