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Turaev-Viro Modules of Satellite knots

Patrick M. Gilmer

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Abstract

Given an oriented knot K in S^3 and a TQFT, Turaev and Viro defined modules somewhat analogous to the Alexander module. We work with the (V_p,Z_p) theories of Blanchet, Habegger, Masbaum and Vogel {BHMV} for p \ge 3, and consider the associated modules. In {G}, we defined modules which also depend on the extra data of a color c which is assigned to a meridian of the knot in the construction of the module. These modules can be used to calculate the quantum invariants of cyclic branched covers of knots and have other uses. Suppose now that S is a satellite knot with companion C, and pattern P. We give formulas for the Turaev-Viro modules for S in terms of the Turaev-Viro modules of C and similar data coming from the pattern P. We compute these invariants explicitly in several examples.

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Given an oriented knot K in S^3 and a TQFT, Turaev and Viro defined modules somewhat analogous to the Alexander module. We work with the (V_p,Z_p) theories of Blanchet, Habegger, Masbaum and Vogel {BHMV} for p \ge 3, and consider the associated modules. In {G}, we defined modules which also depend on the extra data of a color c which is assigned to a meridian of the knot in the construction of the module. These modules can be used to calculate the quantum invariants of cyclic branched covers of knots and have other uses. Suppose now that S is a satellite knot with companion C, and pattern P. We give formulas for the Turaev-Viro modules for S in terms of the Turaev-Viro modules of C and similar data coming from the pattern P. We compute these invariants explicitly in several examples.

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Available abstract

Given an oriented knot K in S^3 and a TQFT, Turaev and Viro defined modules somewhat analogous to the Alexander module. We work with the (V_p,Z_p) theories of Blanchet, Habegger, Masbaum and Vogel {BHMV} for p \ge 3, and consider the associated modules. In {G}, we defined modules which also depend on the extra data of a color c which is assigned to a meridian of the knot in the construction of the module. These modules can be used to calculate the quantum invariants of cyclic branched covers of knots and have other uses. Suppose now that S is a satellite knot with companion C, and pattern P. We give formulas for the Turaev-Viro modules for S in terms of the Turaev-Viro modules of C and similar data coming from the pattern P. We compute these invariants explicitly in several examples.

Key concepts: Topological quantum field theory, Knot (papermaking), Knot theory, Mathematics, Knot invariant, Quantum invariant, Pure mathematics, Algebra over a field

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