Séminaire Lotharingien de Combinatoire, 93B.16 (2025), 12 pp.
Elena S. Hafner, Karola Mészáros and Alexander Vidinas
Polytopal Perspectives on the Alexander Polynomial of Special Alternating Links
Abstract.
The Alexander polynomial (1928) was the first polynomial invariant of
links devised to help distinguish them up to isotopy. In this
abstract, we highlight views of the Alexander polynomials of special
alternating links in terms of polytopes, namely, generalized
permutahedra and root polytopes. Polytopes have been previously used
to study polynomial invariants of special alternating links in the
works of Juhász, K6aacute;lmán, and Rasmussen (2012)
and Kálmán
and Postnikov (2017). We settle Fox's longstanding conjecture of the
trapezoidal property of the Alexander polynomials of alternating links
in the special case of special alternating links using generalized
permutahedra. We also offer a simple explanation of the connection
between the generalized Alexander polynomial of Eulerian graphs
defined by Murasugi and Stoimenow (2003) and root polytopes of
unimodular matrices, building on the works of Li and Postnikov (2013)
and Tóthmérész (2023).
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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