Extensions of Tong-Yang-Ma representation

TL;DR

This paper extends the Tong-Yang-Ma family of braid group representations to string links and welded string links, explores their kernels via linking numbers, and applies the Long-Moody construction to analyze their properties.

Contribution

It introduces new extensions of Tong-Yang-Ma representations to string links, describes their kernels using linking numbers, and applies the Long-Moody construction for further analysis.

Findings

Extended Tong-Yang-Ma representations to string links and welded string links.

Kernels of these representations are characterized by linking numbers.

Applied Long-Moody construction to study properties of the representations.

Abstract

In 1996, Tong, Yang and Ma defined a family of representations of the braid group which have the same dimensions as the (unreduced) Burau representations but are not equivalent. The Burau representation was defined homologically and extended to the string links in several ways. In this paper, using the method of Silver and Williams, we extend the family of the Tong-Yang-Ma representations to the string links and welded string links. Moreover, we show that the kernels of these representations may be described using some linking numbers. Finally, we apply the Long-Moody construction to the Tong-Yang-Ma representations and study its first properties.