Special Issue: Dedicated to 15th Anniversary of the Seminar “Knots and Representation Theory”; Guest Editors: V. O. Manturov, D. P. Ilyutko and I. M. NikonovNo Access

Geometric and homological considerations of local crossings of $n$ -foams

J. Scott Carter

J. Scott Carter

Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA

E-mail Address: carter@southalabama.edu

Search for more papers by this author

https://doi.org/10.1142/S0218216515410072Cited by:0 (Source: Crossref)

Abstract

Foams in all dimensions are defined by being modeled on a dual structure found in an $n$ -dimensional simplex. Their crossings are studied from geometric and homological points of view. A process of defining invariants thereof is outlined. Interesting research level problems are proposed. From considerations of foams, a partial order on certain families of forests is given via Hasse diagrams.

Keywords:

AMSC: 57Q45

References

1. J. C. Baez and L. Langford, Higher-dimensional algebra. IV. 2-tangles, Adv. Math. 180(2) (2003) 705–764. Crossref, Web of Science, Google Scholar
2. J. S. Carter and A. Ishii, A knotted 2-dimensional foam with non-trivial cocycle invariant, preprint (2012), arXiv:1206.4750. Google Scholar
3. J. S. Carter, A. Ishii, M. Saito and K. Tanaka, Homology for quandles with partial group operations, preprint (2015), arXiv:1506.08271. 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, L. H. Kauffman and M. Saito, Structures and diagrammatics of four-dimensional topological lattice field theories, Adv. Math. 146(1) (1999) 39–100. Crossref, Web of Science, Google Scholar
6. J. S. Carter and S. Y. Yang, Twist spinning knotted trivalent graphs, Proc. Amer. Math. Soc. (2015), doi:10.1090/proc/12801. http://www.ams.org/journals/proc/0000-000-00/S0002-9939-2015-12801-8/S002-9939-2015-12801-8.pdf. Google Scholar
7. J. E. Fischer, Jr., 2-categories and 2-knots, Duke Math. J. 75(2) (1994) 493–526. Crossref, Web of Science, Google Scholar
8. A. Ishii, A multiple conjugation quandle and handlebody-knots, to appear in Topology Appl. . Google Scholar
9. A. Ishii and M. Iwakiri, Y. Jang and K. Oshiro, A $G$ -family of quandles and handlebody-knots, Illinois J. Math. 57 (2013) 817–838. Crossref, Web of Science, Google Scholar
10. D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982) 37–65. Crossref, Web of Science, Google Scholar
11. R. Kashaev, On realizations of Pachner moves in 4D, preprint (2015), arXiv:1504.01979. Google Scholar
12. S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982) 78–88. Google Scholar
13. J. Przytycki, Distributivity versus associativity in the homology theory of algebraic structures, Demonstratio Math. 44 (2011) 823–869. Google Scholar