A family of representations is constructed for the braid group $B_n$. The vector spaces on which the braid group acts are defined as the result of identifying the spaces generated by proper colorings of regular trees of degree $3$ with a marked vertex. This identification is done using a family of canonical isomorphisms. The dimensions of the resulting spaces form the sequence of Fibonacci numbers. We then show how the constructed representations can be extended to invariants of unoriented knots and links in a 3-sphere.
Keywords: braid group, representation, knot invariant, Reshetikhin–Turaev type invariant
Received February 10, 2024
Revised July 28, 2024
Accepted August 5, 2024
Funding Agency: This work was supported by the Russian Science Foundation (project no. 23-21-10014, https://rscf.ru/en/project/23-21-10014/).
Philipp Glebovich Korablev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: korablev@csu.ru
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Cite this article as: F.G. Korablev. Fibonacci representations of braid groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 149–169.
