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 = 1 ∪ · · · ∪ n , we require as input an n-tuple (a 1 , . . . , a n ) ∈ Z n of labels, where a j is associated with 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 π 1 (S 3 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 , . . . , a n ), the invariants h N,a (L) vanish whenever L is a split link.

A multivariable Casson–Lin type invariant

“…By construction, h L recovers the invariant of Heusener-Kroll [27] if L is a knot, while Proposition 6.6 shows that h L is locally constant. Note that since we are counting SU(2) representations and not projective SU (2) representations, our invariant h L is distinct from the link invariant constructed by Harper-Saveliev [22] and Boden-Harper [4]. The following paragraphs shall make this difference more concrete.…”

“…Similar invariants have been constructed for links: Harper-Saveliev [22] defined a signed count of a certain type of projective SU(2) representations for 2-component links L = K 1 ∪ K 2 and showed that their invariant coincides with the linking number ± k(K 1 , K 2 ). The sign was later determined by Boden-Herald [5] and the construction was extended to n-component links by Boden-Harper [4]. We also refer to [4] for a construction involving the group SU(n) and to [11,23] for further gauge theoretic developments.…”

“…The sign was later determined by Boden-Herald [5] and the construction was extended to n-component links by Boden-Harper [4]. We also refer to [4] for a construction involving the group SU(n) and to [11,23] for further gauge theoretic developments. The first aim of this article is to produce a multivariable generalization of the Casson-Lin invariant.…”

The SU(2) Casson–Lin invariant of the Hopf link

“…We choose vectors w 1 = (k, 0, 0, 0), w 2 = (0, k, 0, 0), w 3 = (j, 0, 0, 0) to extend this to a basis {u 1 …”

“…The orientation conventions in the definition of the h N,a (L) [1] involve pulling back the orientation from su(2) = T 1 SU (2) by df to obtain a co-orientation for ker df | (i,j,i,j) . With that in mind, we compute the action of df on {w 1 , w 2 , w 3 }, namely, df (w 1 ) = −j, df (w 2 ) = i and df (w 3 ) = k.…”