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Mathematics of Computation

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An enumeration process for racks

Authors: Jim Hoste and Patrick D. Shanahan
Journal: Math. Comp. 88 (2019), 1427-1448
MSC (2010): Primary 20-04; Secondary 57M25
Published electronically: August 31, 2018
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Abstract: Given a presentation for a rack $ \mathcal R$, we define a process which systematically enumerates the elements of $ \mathcal R$. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for $ n$-quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt, Eick, and O'Brien for the Todd-Coxeter process. We prove that the process terminates if and only if $ \mathcal R$ is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot theory.

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Additional Information

Jim Hoste
Affiliation: Department of Mathematics, Pitzer College, 1050 N Mills Avenue, Claremont, California 91711

Patrick D. Shanahan
Affiliation: Department of Mathematics, Loyola Marymount University, UHall 2700, Los Angeles, California 90045

Received by editor(s): July 12, 2017
Received by editor(s) in revised form: February 16, 2018
Published electronically: August 31, 2018
Article copyright: © Copyright 2018 American Mathematical Society