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

Edwin Kitaeff

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arXiv · 2410.23153

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

Edwin Kitaeff

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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.

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BibTeX RIS

Edwin Kitaeff. 2024-10-30. The Gilmer-Masbaum map is not injective on the Kauffman bracket skein module. https://arxiv.org/abs/2410.23153

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