New York Journal of Mathematics
Volume 24 (2018) 293-316


Stephan D. Burton

The determinant and volume of 2-bridge links and alternating 3-braids

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Published: March 8, 2018
Keywords: Hyperbolic knot, knot volume, knot determinant, spanning trees
Subject: 57M25

We examine the conjecture, due to Champanerkar, Kofman, and Purcell, 2015, that \vol[K] < 2 π log \det (K) for alternating hyperbolic links, where \vol[K] = \vol[S3\backslash K] is the hyperbolic volume and \det(K) is the determinant of K. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.


Supported by NSF Grants DMS-1105843 and DMS-1404754.

Author information

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824