Combinatorial identities in the theory of \(\text{SU}(n)\) Casson invariants of knots
Given a fibered knot \(K\) in a closed oriented 3-manifold and \(\alpha \in \text{SU}(n)\) (the special unitary group), the Casson invariant of \(K\), denoted by \(\lambda_{n, \alpha}(K)\), is an integer which can be viewed as an algebraic-topological count of the number of characters of \(\text{SU}(n)\) representations of the knot group which take a longitude into a conjugacy class of \(\alpha\) [see \textit{H. U. Boden}, Invariants of fibred knots from moduli. Geometric topology. 1993 Georgia international topology conference, August 2--13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2 (pt. 1), 259--267 (1997; Zbl 0889.57006), Trans. Am. Math. Soc. 352, 3149--3187 (2000; Zbl 0945.57008), Topology 32, 121--144 (1993; Zbl 0791.57005)]. In [Trans. Am. Math. Soc. 352, 3149--3187 (2000; Zbl 0945.57008)], H. U. Boden and A. Nicas make use of Zagier's summation formula [see Isr. Math. Conf. Proc. 9, 445--462 (1996; Zbl 0854.14020)] applied to Atiyah-Bott recursion [see Philos. Trans. R. Soc. Lond., A 308, 523--615 (1983; Zbl 0509.14014)] in order to express \(\lambda_{n, \alpha}(K)\) by means of explicit linear combinations of sums of a well-defined form; in the same paper, they compute the value of these sums - depending on an integer \(d\) - for \(d=1\) and make a conjecture about it in the general case. The present paper gives an explicit evaluation of the above sum in the case \(d=2\); as a consequence, Boden and Nicas' conjecture is proved to be true for \(d=2\), too. Moreover, the techniques developed in the present paper are expected to be of use also in order to treat the general case.
scientific article; zbMATH DE number 5074506