Vol. 3, No. 2, 2010

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The Gram determinant for plane curves

Józef H. Przytycki and Xiaoqi Zhu

Vol. 3 (2010), No. 2, 149–169
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