## 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.

### MSC:

 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
Full Text:

### References:

 [1] H Abchir, TQFT invariants at infinity for the Whitehead manifold, Ser. Knots Everything 24, World Sci. Publ., River Edge, NJ (2000) 1 · Zbl 0978.57026 [2] C Blanchet, N Habegger, G Masbaum, P Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992) 685 · Zbl 0771.57004 [3] C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883 · Zbl 0887.57009 [4] K Habiro, On the colored Jones polynomials of some simple links, S\Burikaisekikenky\Busho K\Boky\Buroku (2000) 34 · Zbl 0969.57503 [5] K Habiro, On the quantum $$\mathrm sl_2$$ invariants of knots and integral homology spheres, Geom. Topol. Monogr. 4, Geom. Topol. Publ., Coventry (2002) 55 · Zbl 1040.57010 [6] K Habiro, in preparation [7] R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 · Zbl 0876.57007 [8] L H Kauffman, S L Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994) · Zbl 0821.57003 [9] T T Q Le, Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion (2003) 125 · Zbl 1020.57002 [10] T Q T Le, in preparation [11] G Masbaum, P Vogel, 3-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994) 361 · Zbl 0838.57007 [12] H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85 · Zbl 0983.57009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.