The SU(N) Casson-Lin invariants for links

Abstract

We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L={\ell }_1\cup \cdots \cup {\ell }_n$, we require as input an $n$-tuple $(a_1,\dots ,a_n)\in {\mathbb{Z}}^n$ of labels, where $a_j$ is associated with ${\ell }_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(L)$, gives an algebraic count of certain projective $SU(N)$ representations of the link group ${\pi }_1(S^3\setminus L)$, and the family $h_{N,a}$ of link invariants gives a natural extension of the $SU(2)$ Casson-Lin invariant, which was defined for knots by X.-S. Lin and for 2-component links by Harper and Saveliev. We compute the invariants for the Hopf link and more generally for chain links, and we show that, under mild conditions on the labels $(a_1,\dots ,a_n)$, the invariants $h_{N,a}(L)$ vanish whenever $L$ is a split link.

24 pages, 3 figures. Revised to clarify relationship between geometric braids and the automorphisms they induce on the free group

References in corpus (1)

• Knot homology groups from instantons

Cited by in corpus (1)

• The Benard-Conway invariant of two-component links