Burau Maps and Twisted Alexander Polynomials

Abstract

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

A multivariable Casson–Lin type invariant

“…In this subsection, we briefly review colored braids and discuss the action of the colored braid groups on SU(2) n . References for colored braids include [12,39], while discussions of the action of the braid group B n on SU(2) n include [26,27,35,36].…”

“…In this subsection, we recall the definition of the colored Gassner matrices and of the reduced colored Gassner matrices which are multivariable generalizations of the (reduced) Burau matrices. Although references include [10,14,32], our conventions are closest to those of [12].…”

“…Since L has 2 components and linking number 1, the Torres formula (which reads ∆ K1∪K2 (t 1 , 1) . = (t 12 1 − 1)/(t 1 − 1)∆ K1 (t 1 ), where 12 = k(K 1 , K 2 )) shows that |∆ L (1, 1)| = 1. Thus ∆ L is not identically zero and therefore the multivariable Casson-Lin invariant h L (α) is well defined whenever ∆ L (ω) = 0 for all ω ∈ S(α).…”

“…and[12, Section 3 (c)], we deal with the reduced colored Gassner matrices. Instead of working with the free generators x 1 , x 2 .…”

The Long-Moody construction and twisted Alexander invariants

“…As a generalization of the Alexander polynomial, Wada [11] and Lin [7] independently defined the twisted Alexander invariant which is an invariant of a given link and a representation of the fundamental group of the link complement. Conway [4] introduced the twisted Burau map B ρ : B n − GL nk (R[t ±1 ]), where R is a commutative ring and ρ : F n − GL k (R) is a representation of the free group F n . This is defined as a homomorphism of the twisted homology of the n-punctured disk, and generally not a representation.…”