Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2105.02037

New structures for colored HOMFLY-PT invariants

Shengmao Zhu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by H. Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then, we apply our results to refine the mathematical structures appearing in the Labastida-Mari\~no-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another application of the strong integrality, we obtain that the $q=1$ and $a=1$ specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. We find that a conjectural formula for the colored Alexander polynomial which is the $a=1$ specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for frame knot holds.

Explore related subjects

math.GTmath.QA

Keep this discovery

BibTeXRIS

Shengmao Zhu. 2021-05-05. New structures for colored HOMFLY-PT invariants. https://arxiv.org/abs/2105.02037

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↗