[Paper Review] Turaev-Viro Modules of Satellite knots

Patrick M. Gilmer|arXiv (Cornell University)|Oct 11, 1996

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper establishes explicit formulas for Turaev-Viro modules of satellite knots in terms of the modules of their companion and pattern knots within the (V_p, Z_p) TQFT framework for p ≥ 3. It extends the Turaev-Viro construction by incorporating meridian color data, enabling computation of quantum invariants for cyclic branched covers and providing explicit calculations in multiple examples, thus generalizing module structures to satellite knot theory.

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.

Motivation & Objective

• To extend the Turaev-Viro module construction to satellite knots using the (V_p, Z_p) TQFT for p ≥ 3.
• To incorporate meridian coloring data into the module construction, enabling computation of quantum invariants of cyclic branched covers.
• To derive explicit formulas expressing the Turaev-Viro module of a satellite knot in terms of the modules of its companion and pattern.
• To compute these invariants explicitly in several nontrivial examples to demonstrate the utility of the formulas.

Proposed method

• Utilizes the (V_p, Z_p) TQFT of Blanchet, Habegger, Masbaum, and Vogel for prime p ≥ 3.
• Applies the extended Turaev-Viro module construction that depends on a color c assigned to a meridian of the knot.
• Expresses the Turaev-Viro module of a satellite knot S with companion C and pattern P as a tensor product or induced module construction involving the modules of C and P.
• Employs diagrammatic techniques and state-sum models from TQFT to compute the module structure from the satellite decomposition.
• Uses the satellite structure to decompose the state-sum over the pattern and companion, leveraging the gluing properties of TQFTs.
• Validates the formulas through explicit computations in multiple examples, including specific satellite knots.

Experimental results

Research questions

• RQ1 How can the Turaev-Viro module of a satellite knot be expressed in terms of the modules of its companion and pattern knots?
• RQ2 What role does the meridian coloring c play in the construction of Turaev-Viro modules for satellite knots?
• RQ3 Can the quantum invariants of cyclic branched covers of satellite knots be computed using the extended Turaev-Viro module framework?
• RQ4 What are the structural properties of Turaev-Viro modules under satellite operations in the (V_p, Z_p) TQFT?
• RQ5 How do the module formulas behave under different choices of p and meridian colors in concrete examples?

Key findings

• The Turaev-Viro module of a satellite knot S with companion C and pattern P is explicitly computed as a module induced from the tensor product of the modules of C and P, incorporating the meridian color c.
• The formulas allow direct computation of quantum invariants for cyclic branched covers of satellite knots, extending previous results to a broader class of knots.
• Explicit computations are provided for several examples, demonstrating the practical utility of the derived formulas.
• The construction preserves the TQFT structure and respects the satellite decomposition, showing consistency with topological expectations.
• The inclusion of meridian coloring c enriches the module structure and enables finer invariants than the standard Turaev-Viro modules.
• The results generalize the Turaev-Viro module framework to satellite knots, offering a systematic method for computing invariants in this setting.