Research ArticlesNo Access

Quandle coloring and cocycle invariants of composite knots and abelian extensions

W. Edwin Clark

Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA

Search for more papers by this author

Masahico Saito

http://orcid.org/0000-0003-0795-8548

Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA

Search for more papers by this author

, and

Leandro Vendramin

Departamento de Mathemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina

Search for more papers by this author

https://doi.org/10.1142/S0218216516500243Cited by:10 (Source: Crossref)

Abstract

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.

Keywords:

AMSC: 57M25

References

1. N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003) 177–243. Crossref, Web of Science, Google Scholar
2. J. S. Birman and W. Menasco, Studying links via closed braids IV: Composite links and split links, Invent. Math. 102 (1990) 115–139. Crossref, Web of Science, Google Scholar
3. G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics, 2nd edn. Vol. 5 (Walter de Gruyter & Co., Berlin, 2003). Google Scholar
4. J. S. Carter, M. Elhamdadi, M. A. Nikiforou and M. Saito, Extensions of quandles and cocycle knot invariants, J. Knot Theory Ramifications 12 (2003) 725–738. Link, Web of Science, Google Scholar
5. J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003) 3947–3989. Crossref, Web of Science, Google Scholar
6. J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito, Computations of quandle cocycle invariants of knotted curves and surfaces, Adv. Math. 157 (2001) 36–94. Crossref, Web of Science, Google Scholar
7. J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito, Quandle homology groups, their Betti numbers, and virtual knots, J. Pure Appl. Algebra 157(2–3) (2001) 135–155. Crossref, Web of Science, Google Scholar
8. J. S. Carter, S. Kamada and M. Saito, Surfaces in $4$ -space, Encyclopaedia of Mathematical Sciences, Vol. 142 (Springer Verlag, 2004). Crossref, Google Scholar
9. J. S. Carter, S. Kamada and M. Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001) 345–386. Link, Web of Science, Google Scholar
10. J. S. Carter, M. Saito and S. Satoh, Ribbon concordance of surface-knots via quandle cocycle invariants, J. Aust. Math. Soc. 80 (2006) 131–147. Crossref, Web of Science, Google Scholar
11. W. E. Clark, M. Elhamdadi, M. Saito and T. Yeatman, Quandle colorings of knots and applications, J. Knot Theory Ramifications 23 (2014) 1450035, 29 pp. Link, Web of Science, Google Scholar
12. W. E. Clark and T. Yeatman, Properties of Rig quandles, http://math.usf.edu/ $\sim$ saito/QuandleColor/characterization.pdf. Google Scholar
13. J. H. Conway, An Enumeration of Knots and Links, and Some of Their Algebraic Properties, in Computational Problems in Abstract Algebra, ed. J. Leech (Pergamon Press, Oxford, England, 1970), pp. 329–358. Crossref, Google Scholar
14. J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.edu/ $\sim$ knotinfo, May 26, 2011. Google Scholar
15. F. J. B. J. Clauwens, Small connected quandles, http://front.math.ucdavis.edu/1011.2456. Google Scholar
16. M. Eisermann, Homological characterization of the unknot, J. Pure Appl. Algebra 177(2) (2003) 131–157. Crossref, Web of Science, Google Scholar
17. M. Elhamdadi, J. MacQuarrie and R. Restrepo, Automorphism groups of quandles, J. Algebra Appl. 11 (2012) 1250008, 9 pp. Link, Web of Science, Google Scholar
18. R. Fenn and C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992) 343–406. Link, Google Scholar
19. R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995) 321–356. Crossref, Web of Science, Google Scholar
20. J. Hempel, Residual finiteness for 3-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984), pp. 379–396; Ann. of Math. Stud., Vol. 111 (Princeton Univ. Press, Princeton, NJ, 1987). Google Scholar
21. D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37–65. Crossref, Web of Science, Google Scholar
22. L. N. Litherland and S. Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178 (2003) 187–202. Crossref, Web of Science, Google Scholar
23. V. Manturov, Knot Theory (Chapman & Hall/CRC, Boca Raton, FL, 2004). Crossref, Google Scholar
24. S. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) (Russian) 119(161) (1982) 78–88 (160). Google Scholar
25. T. Mochizuki, The third cohomology groups of dihedral quandles, J. Knot Theory Ramifications 20 (2011) 1041–1057. Link, Web of Science, Google Scholar
26. M. Niebrzydowski and J. H. Przytycki, Homology of dihedral quandles, J. Pure Appl. Algebra 213 (2009) 742–755. Crossref, Web of Science, Google Scholar
27. T. Nosaka, On homotopy groups of quandle spaces and the quandle homotopy invariant of links, Topology Appl. 158 (2011) 996–1011. Crossref, Web of Science, Google Scholar
28. C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. Math. 66(2) (1957) 1–26. Crossref, Web of Science, Google Scholar
29. J. H. Przytycki, $3$ -colorings and other elementary invariants of knots, Knot Theory (Warsaw, 1995), pp. 275–295; Banach Center Publ., 42, Polish Acad. Sci., Warsaw (1998). Google Scholar
30. J. H. Przytycki, From 3-moves to Lagrangian tangles and cubic skein modules, in Advances in Topological Quantum Field Theory, Proceedings of the NATO ARW on New Techniques in Topological Quantum Field Theory, ed. John M. Bryden (2004), pp. 71–125. Google Scholar
31. M. Takasaki, Abstraction of symmetric transformation, Tohoku Math. J. (in Japanese), 49 (1942/3) 145–207. Google Scholar
32. W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982) 357–381. Crossref, Web of Science, Google Scholar
33. L. Vendramin, Rig — A GAP package for racks and quandles. Available at http://github.com/vendramin/rig/. Google Scholar
34. L. Vendramin, On the classification of quandles of low order, J. Knot Theory Ramifications 21(9) (2012) 1250088. Link, Web of Science, Google Scholar