Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1801.00580

A formula for the action of Dehn twists on HOMFLY-PT skein modules and its applications

Shunsuke Tsuji
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface. As an application of the formula to mapping class group, we give an embedding from the Torelli group of a surface $\Sigma_{g,1}$ of genus $g$ with non-empty connected boundary into the completed HOMFLY-PT type skein algebra. As an application of the formula to integral homology $3$-spheres, we construct an invariant $z(M) \in \mathbb{Q} [\rho ] [[h]]$ for an integral homology $3$-sphere $M$. The invariant $z(M) \mod (h^{n+1})$ is a finite type invariant of order $n$.

Explore related subjects

math.GT

Keep this discovery

BibTeXRIS

Shunsuke Tsuji. 2018-01-02. A formula for the action of Dehn twists on HOMFLY-PT skein modules and its applications. https://arxiv.org/abs/1801.00580

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗
arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗