A contribution of a U(1)-reducible connection to quantum invariants of links I: R-matrix and Burau representation

Burau representation and random walk on string links

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. (This is an extensively revised version of the first author's original paper titled "Burau repr…

Notes on link homology

This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.

A contribution of a U(1)-reducible connection to quantum invariants of links II: Links in rational homology spheres

We extend the definition of the U(1)-reducible connection contribution to the case of the Witten-Reshetikhin-Turaev invariant of a link in a rational homology sphere. We prove that, similarly ot the case of a link in S^3, this contribution is a formal power series in powers of q-1, whose coefficients are rational functions of q^{color}, their denominators being the powers of the Alexander-Conway polynomial. The coefficients of the polynomials in numerators are rational numbers, the bounds on their denominators are established with the help of the theorem proved by T. Ohtsuki in Appendix 2. Sim…

Holonomy invariants of links and nonabelian Reidemeister torsion

We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $\mathcal{U}_q(\mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $\operatorname{SL}_2(\mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their in…

Knot Floer 同调、Burau 表示与量子 $\mathfrak{gl}(1 \vert 1)$

Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$

本文通过 Fox 微积分建立 Burau 表示与 Knot Floer 同调的明确联系，定义范畴化行列式的不变量，并利用量子 $\mathfrak{gl}(1 \vert 1)$ 证明其恢复辫表示。

Quantum theory of non-abelian differential forms and link polynomials

A topological quantum field theory of non-abelian differential forms is investigated from the point of view of its possible applications to description of polynomial invariants of higher-dimensional two-component links. A path-integral representation of the partition function of the theory, which is a highly on-shell reducible system, is obtained in the framework of the antibracket-antifield formalism of Batalin and Vilkovisky. The quasi-monodromy matrix, giving rise to corresponding skein relations, is formally derived in a manifestly covariant non-perturbative manner.