A combinatorial matrix in 3-manifold theory

Abstract

In this paper we study the matrix A(n) which was defined by W. B. R. Lickorish [3]. We prove a result required by Lickorish which completes his topological and combinatorial approach to the 3-manifold invariants of Witten-Reshetikhin-Turaev [4], [5]. This matrix arises from a pairing on a set of geometric configurations. These are the configurations of n nonintersecting arcs in the disk with 2n specified boundary points. There are Cn such configurations where Cn is the nth Catalan number so the matrix increases in size very rapidly. The Catalan numbers were discovered by Euler who considered the ways to partition a polygon into triangles [1]. These two counting problems correspond naturally by considering restricted sequences. The matrix has entries in Z[δ]. Lickorish needed that det A(n) = 0 if δ = ±2 cos j^γ. We find a recursive formula for det A(n) and show that all the roots are of the form 2 cos -£fγ for 1 < m < n and 1 < k < m and verify the result. Using this formula, we derive a simple rule that allows one to recursively compute detA(n) by generating all of its factors. There have been three approaches to study polynomial invariants of classical links: the topological and combinatorial approach considered by Kauffman, Lickorish and many other topologists; the study of quantized Yang-Baxter equations and related Lie algebras by Reshetikhin and Turaev; and the study of subfactors and traces of von Neumann and Hecke algebras by Jones. We took a topological and combinatorial viewpoint. The authors have been informed that the essential result needed by Lickorish could have been obtained by pursuing the two other approaches.