Splitting maps in link Floer homology and integer points in permutahedra

TL;DR

This paper explores the skein exact sequence in link Floer homology, relating splitting maps of torus links to permutahedron integer points and computing Floer homology for certain links.

Contribution

It introduces a new connection between splitting maps in link Floer homology and combinatorial geometry of permutahedra, expanding understanding of link invariants.

Findings

Splitting maps for $T(n,n)$ links correspond to integer points in permutahedra.

Computed link Floer homology for an $n$-component unlink in $S^{1} imes S^{2}$.

Established relations between skein exact sequences and splitting number analysis.

Abstract

In this paper, we study the skein exact sequence for links via the exact surgery triangle of link Floer homology and compare it with other skein exact sequences given by Ozsv\'ath and Szab\'o. As an application, we use the skein exact sequence to study the splitting number and splitting maps for links. In particular, we associate the splitting maps for the torus link $T(n, n)$ to integer points in the $(n-1)$-dimensional permutahedron, and obtain the link Floer homology of an $n$-component homology nontrivial unlink in $S^{1}\times S^{2}$.