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

Abstract

Abstract. We compute the SU (2) Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.The Casson-Lin knot invariant was defined by Lin [4], and then extended to an invariant h 2 (L) of 2-component links by Harper and Saveliev [2], who showed that h 2 (L) = ±lk( 1 , 2 ), the linking number of L = 1 ∪ 2 , up to an overall sign. The purpose of this note is to determine the sign in that formula, establishing the following.is a signed count of certain projective SU (2) representations of the link group. This paper is devoted to proving Theorem 1. The proof of Proposition 5.7 in [2] shows that the sign in the above formula is independent of L, and thus Theorem 1 will follow from a single computation.To that end, we will determine the Casson-Lin invariant for the right-handed Hopf link. Since there is just one irreducible projective SU (2) representation of the link group, up to conjugation, it suffices to determine the sign associated to this one point.We identifywith the group of unit quaternions and consider the conjugacy classof purely imaginary unit quaternions. Notice that C i is diffeomorphic to S 2 and coincides with the set of SU (2) matrices of trace zero.Let L be an oriented link in S 3 , represented as the closure of an n-strand braid σ ∈ B n . We follow Conventions 1.13 on p.17 of [3] for writing geometric braids σ as words in the standard generators σ 1 , . . . , σ n−1 . In particular, braids are oriented from top to bottom and σ i denotes a right-handed crossing in which the (i + 1)-st

A multivariable Casson–Lin type invariant

“…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.…”