Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2102.03487

Values of the $\mathfrak{sl}_2$ weight system on complete bipartite graphs

P. Filippova
View original source ↗Read source document ↗Export metadata ↓

Abstract

A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra $\mathfrak{sl}_2$, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. In the present paper we give a formula for the values of the $\mathfrak{sl}_2$ weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts. Our main computational tool is the Chmutov--Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the $\mathfrak{sl}_2$ weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S.K.Lando on the values of the weight system $\mathfrak{sl}_2$ at the projections of chord diagrams on the space of primitive elements.

Explore related subjects

math.GTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

P. Filippova. 2021-02-06. Values of the $\mathfrak{sl}_2$ weight system on complete bipartite graphs. https://doi.org/10.1134/s0016266320030065

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

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗