An enumeration process for racks
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- by Jim Hoste and Patrick D. Shanahan HTML | PDF
- Math. Comp. 88 (2019), 1427-1448 Request permission
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.References
- Mohamed Elhamdadi and Sam Nelson, Quandles—an introduction to the algebra of knots, Student Mathematical Library, vol. 74, American Mathematical Society, Providence, RI, 2015. MR 3379534, DOI 10.1090/stml/074
- Roger Fenn and Colin Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), no. 4, 343–406. MR 1194995, DOI 10.1142/S0218216592000203
- Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2129747, DOI 10.1201/9781420035216
- Jim Hoste and Patrick D. Shanahan, Involutory quandles of $(2,2,r)$-Montesinos links, J. Knot Theory Ramifications 26 (2017), no. 3, 1741003, 19. MR 3627703, DOI 10.1142/S0218216517410036
- Jim Hoste and Patrick D. Shanahan, Links with finite $n$-quandles, Algebr. Geom. Topol. 17 (2017), no. 5, 2807–2823. MR 3704243, DOI 10.2140/agt.2017.17.2807
- D. L. Johnson, Presentations of groups, 2nd ed., London Mathematical Society Student Texts, vol. 15, Cambridge University Press, Cambridge, 1997. MR 1472735, DOI 10.1017/CBO9781139168410
- David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65. MR 638121, DOI 10.1016/0022-4049(82)90077-9
- David Edward Joyce, AN ALGEBRAIC APPROACH TO SYMMETRY WITH APPLICATIONS TO KNOT THEORY, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–University of Pennsylvania. MR 2628474
- S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78–88, 160 (Russian). MR 672410
- J. A. Todd and H. S. M. Coxeter, A practical method for enumerating cosets of a finite abstract group, Proceedings of the Edinburgh Mathematical Society, Series II, 5, pp. 26–34, (1936).
- J. N. Ward, A note on the Todd-Coxeter algorithm, Group theory (Proc. Miniconf., Australian Nat. Univ., Canberra, 1975) Lecture Notes in Math., Vol. 573, Springer, Berlin, 1977, pp. 126–129. MR 0447411
- Steven Karl Winker, QUANDLES, KNOT INVARIANTS, AND THE N-FOLD BRANCHED COVER, ProQuest LLC, Ann Arbor, MI, 1984. Thesis (Ph.D.)–University of Illinois at Chicago. MR 2634013
Additional Information
- Jim Hoste
- Affiliation: Department of Mathematics, Pitzer College, 1050 N Mills Avenue, Claremont, California 91711
- MR Author ID: 88610
- Email: jhoste@pitzer.edu
- Patrick D. Shanahan
- Affiliation: Department of Mathematics, Loyola Marymount University, UHall 2700, Los Angeles, California 90045
- MR Author ID: 537475
- Email: pshanahan@lmu.edu
- Received by editor(s): July 12, 2017
- Received by editor(s) in revised form: February 16, 2018
- Published electronically: August 31, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1427-1448
- MSC (2010): Primary 20-04; Secondary 57M25
- DOI: https://doi.org/10.1090/mcom/3374
- MathSciNet review: 3904151