The Gilmer-Masbaum map is not injective on the Kauffman bracket skein module

TL;DR

This paper investigates the injectivity of the Gilmer-Masbaum map on the Kauffman bracket skein module, revealing non-injectivity in specific cases and providing a basis for certain mapping tori.

Contribution

It demonstrates the non-injectivity of the Gilmer-Masbaum map on specific homology classes and constructs a basis for the skein module of certain mapping tori.

Findings

The map is not injective on certain homology classes.

Provides a basis for the skein module of the mapping torus of a power of the Dehn twist.

Answers a question posed by Gilmer and Masbaum.

Abstract

Gilmer and Masbaum use Witten-Reshetikhin-Turaev (WRT) invariants to define a map from the Kauffman bracket skein module to a set of complex-valued functions defined on roots of unity in order to provide a lower bound for its dimension. We compute the image of the evaluation map for a family of mapping tori of the 2-torus and find that the restriction of the map to certain homology classes is not injective. This provides an answer to a question posed by Gilmer and Masbaum in the latter paper. Moreover, we give a basis for the Kauffman bracket skein module in the case of the mapping torus of a power of the Dehn twist along the meridian of the 2-torus.