Quantum Topology

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Volume 8, Issue 2, 2017, pp. 205–248
DOI: 10.4171/QT/89

Published online: 2017-05-29

Root polytopes, parking functions, and the HOMFLY polynomial

Tamás Kálmán[1] and Hitoshi Murakami[2]

(1) Tokyo Institute of Technology, Japan
(2) Tohoku University, Sendai, Japan

We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of parking functions for the planar dual of the Seifert graph. These observations yield formulas for the maximal $z$-degree part of the HOMFLY polynomial of an arbitrary homogeneous link as well. Our result is part of a program aimed at reading HOMFLY coefficients out of Floer homology.

Keywords: Alternating link, HOMFLY polynomial, Seifert graph, root polytope, $h$-vector, parking function

Kálmán Tamás, Murakami Hitoshi: Root polytopes, parking functions, and the HOMFLY polynomial. Quantum Topol. 8 (2017), 205-248. doi: 10.4171/QT/89