Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 002, 12 pages      arXiv:2101.11708

A Cable Knot and BPS-Series

John Chae
Department of Mathematics, Univeristy of California Davis, Davis, USA

Received August 03, 2022, in final form January 05, 2023; Published online January 13, 2023

A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we observe a relation between the series invariant of the cable knot and the series invariant of the figure eight knot. This relation provides an alternative simple method of finding the former series invariant.

Key words: knot complement; quantum invariant; $q$-series; Chern-Simons theory; categorification.

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