The Gram determinant for plane curves

Przytycki, Józef; Zhu, Xiaoqi

doi:10.2140/involve.2010.3.149

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The Gram determinant for plane curves

Józef H. Przytycki and Xiaoqi Zhu

Vol. 3 (2010), No. 2, 149–169

DOI: 10.2140/involve.2010.3.149

Abstract

We investigate the Gram determinant of the pairing arising from curves in a planar surface, with a focus on the disk with two holes. We prove that the determinant based on $n - 1$ curves divides the determinant based on $n$ curves. Motivated by the work on Gram determinants based on curves in a disk and curves in an annulus (Temperley–Lieb algebra of type $A$ and $B$ , respectively), we calculate several examples of the Gram determinant based on curves in a disk with two holes, and advance conjectures on the complete factorization of Gram determinants.

Keywords

Gram determinants, planar curves, noncrossing partitions, chromatic joins, Temperley–Lieb

Mathematical Subject Classification 2000

Primary: 05A99, 57M99

Milestones

Received: 30 November 2008

Revised: 27 April 2010

Accepted: 3 June 2010

Published: 11 August 2010

Communicated by Kenneth S. Berenhaut

Authors

Józef H. Przytycki
Department of Mathematics The George Washington University Washington, DC 20052 United States

Xiaoqi Zhu
Department of Applied Mathematics Harvard University Cambridge, MA 02138 United States

Vol. 3, No. 2, 2010