Vol. 12, No. 8, 2019

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Unoriented links and the Jones polynomial

Sandy Ganzell, Janet Huffman, Leslie Mavrakis, Kaitlin Tademy and Griffin Walker

Vol. 12 (2019), No. 8, 1357–1367
Abstract

The Jones polynomial is an invariant of oriented links with n 1 components. When n = 1, the choice of orientation does not affect the polynomial, but for n > 1, changing orientations of some (but not all) components can change the polynomial. Here we define a version of the Jones polynomial that is an invariant of unoriented links; i.e., changing orientation of any sublink does not affect the polynomial. This invariant shares some, but not all, of the properties of the Jones polynomial.

The construction of this invariant also reveals new information about the original Jones polynomial. Specifically, we show that the Jones polynomial of a knot is never the product of a nontrivial monomial with another Jones polynomial.

Keywords
Jones polynomial, unoriented link
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Milestones
Received: 10 April 2019
Revised: 19 June 2019
Accepted: 6 July 2019
Published: 25 October 2019

Communicated by Joel Foisy
Authors
Sandy Ganzell
Department of Mathematics and Computer Science
St. Mary’s College of Maryland
St. Mary’s City, MD
United States
Janet Huffman
University of Kentucky
Lexington, KY
United States
Leslie Mavrakis
University of California
Santa Barbara, CA
United States
Kaitlin Tademy
University of Nebraska
Lincoln, NE
United States
Griffin Walker
Wheaton College
Wheaton, IL
United States