No Access

Biquandle cohomology and state-sum invariants of links and surface-links

Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: skamada@sci.osaka-cu.ac.jp

Osaka City University Advanced Mathematical Institute, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: kawauchi@sci.osaka-cu.ac.jp

Search for more papers by this author

Jieon Kim

Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea

E-mail Address: jieonkim@pusan.ac.kr

Search for more papers by this author

, and

Sang Youl Lee

Department of Mathematics, Pusan National University, Busan 46241, Republic of Korea

E-mail Address: sangyoul@pusan.ac.kr

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218216518430162Cited by:3 (Source: Crossref)

This article is part of the issue:

Special Issue — Self-Distributive System and Quandle (Co)Homology Theory in Algebra and Low-Dimensional Topology; Guest Editors: Y. Bae, S. Kamada, A. Kawauchi and S. Y. Lee

Abstract

In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams. We also develop the shadow (co)homology theory of biquandles and construct the shadow biquandle cocycle invariants for oriented surface-links.

Keywords:

AMSC: 57M25, 57M27

References

1. S. Ashihara, Calculating the fundamental biquandles of surface-links from their ch-diagrams, J. Knot Theory Ramifications 21(10) (2012) 1250102. Link, Web of Science, Google Scholar
2. T. Carrell, The Surface Biquandle (Pomona College, 2009). Google Scholar
3. J. S. Carter, M. Elhamdadi and M. Saito, Homology Theory for the Set-Theoretic Yang-Baxter Equation and Knot Invariants from Generalizations of Quandles, Fund. Math. 184 (2004) 31–54. Crossref, Web of Science, Google Scholar
4. 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
5. J. S. Carter, S. Kamada, M. Saito, Geometric interpretations of quandle homology, J. Knot Theory Ramifications 10 (2001) 345–386. Link, Web of Science, Google Scholar
6. J. S. Carter and M. Saito, Knotted surfaces and their diagrams, in Mathematical Surveys and Monographs, Vol. 55 (American Mathematical Society, Providence, RI, 1998). Google Scholar
7. R. Fenn and C. Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992) 343–406. Link, Google Scholar
8. R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3(4) (1995) 321–356. Crossref, Web of Science, Google Scholar
9. D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37–65. Crossref, Web of Science, Google Scholar
10. S. Kamada, J. Kim and S. Y. Lee, Computations of quandle cocycle invariants of surface-links using marked graph diagrams, J. Knot Theory Ramifications 24(10) (2015) 1540010. Link, Web of Science, Google Scholar
11. L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras, and generalized Alexander polynomial for virtual links, Contemp. Math., 318 (2003) 113–140. Crossref, Google Scholar
12. A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four-space, I Normal forms, Math. Sem. Notes Kobe Univ. 10 (1982) 75–125. Google Scholar
13. C. Kearton and V. Kurlin, All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron, Algebr. Geom. Topol. 8 (2008) 1223–1247. Crossref, Web of Science, Google Scholar
14. J. Kim, Y. Joung and S. Y. Lee, On the Alexander biquandles of oriented surface-links via marked graph diagrams, J. Knot Theory Ramifications 23(7) (2014) 1460007. Link, Web of Science, Google Scholar
15. J. Kim, Y. Joung and S. Y. Lee, On generating sets of Yoshikawa moves for marked graph diagrams of surface-links, J. Knot Theory Ramifications 24(4) (2015) 1550018. Link, Web of Science, Google Scholar
16. S. Y. Lee, Shadow biquandle cohomology and state-sum invariants of link cobordisms, preprint (2018). Google Scholar
17. S. J. Lomonaco, The homotopy groups of knots I. How to compute the algebraic $2$ -type, Pacific J. Math. 95 (1981) 349–390. Crossref, Web of Science, Google Scholar
18. S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119 (161) (1982) 78–88 (in Russian); Math. USSR-Sb. 47 (1984) 73–83 (in English). Google Scholar
19. S. Nelson and J. Rosenfield, Bikei homology, Homotopy & Appl. 19 (2017) 23–35. Crossref, Web of Science, Google Scholar
20. S. Nelson and J. Vo, Matrices and finite biquandles, Homology, Homotopy Appl. 8(2) (2006) 51–73. Crossref, Web of Science, Google Scholar
21. M. Polyak, Minimal generating sets of Reidemeister moves, Quantum Topol. 1(4) (2010) 399–411. Crossref, Google Scholar
22. D. Roseman, Reidemeister-type moves for surfaces in four dimensional space, in Knot Theory Vol. 42 (Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1998) pp. 347–380. Crossref, Google Scholar
23. F. J. Swenton, On a calculus for $2$ -knots and surfaces in $4$ -space, J. Knot Theory Ramifications 10 (2001) 1133–1141. Link, Web of Science, Google Scholar
24. K. Yoshikawa, An enumeration of surfaces in four-space, Osaka J. Math. 31 (1994) 497–522. Web of Science, Google Scholar