Research ArticleNo Access

Minimal generating set of planar moves for surfaces embedded in the four-space

Michał Jabłonowski

Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland

E-mail Address: michal.jablonowski@gmail.com

https://doi.org/10.1142/S0218216521500620Cited by:0 (Source: Crossref)

Abstract

We derive a minimal generating set of planar moves for diagrams of surfaces embedded in the four-space. These diagrams appear as the bonded classical unlink diagrams.

Keywords:

AMSC: 57K45

References

1. J. S. Carter, S. Kamada and M. Saito , Surfaces in 4-Space, Encyclopaedia of Mathematical Sciences: Low-Dimensional Topology, III, Vol. 142 (Springer-Verlag, Berlin, 2004). Crossref, Google Scholar
2. R. H. Fox , A quick trip through knot theory, in Topology of 3-Manifolds and Related Topics (Prentice-Hall, 1962), pp. 120–167. Google Scholar
3. B. Gabrovšek , An invariant for colored bonded knots, Stud. Appl. Math. 146 (2021) 586–604. Crossref, Web of Science, Google Scholar
4. M. Jabłonowski , On a surface singular braid monoid, Topology Appl. 160 (2013) 1773–1780. Crossref, Google Scholar
5. M. Jabłonowski , On a banded link presentation of knotted surfaces, J. Knot Theory Ramifications 25 (2016) 1640004. Link, Web of Science, Google Scholar
6. M. Jabłonowski , Minimal hard surface-unlink and classical unlink diagrams, J. Knot Theory Ramifications 28 (2019) 1940002. Link, Web of Science, Google Scholar
7. M. Jabłonowski , Independence of Yoshikawa eighth move and a minimal generating set of band moves, Fund. Math. 251 (2020) 183–193. Crossref, Web of Science, Google Scholar
8. Y. Joung, J. Kim and S. Y. Lee , Ideal coset invariants for surface-links in $ℝ^{4}$ , J. Knot Theory Ramifications 22 (2013) 1350052. Link, Web of Science, Google Scholar
9. Y. Joung, J. Kim and S. Y. Lee , On generating sets of Yoshikawa moves for marked graph diagrams of surface-links, J. Knot Theory Ramifications 24 (2015) 1550018. Link, Web of Science, Google Scholar
10. S. Kamada , Non-orientable surfaces in $4$ -space, Osaka J. Math. 26 (1989) 367–385. Web of Science, Google Scholar
11. S. Kamada , Surface-Knots in 4-Space, Springer Monographs in Mathematics (Springer, 2017). Crossref, Google Scholar
12. L. H. Kauffman , Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311(2) (1989) 697–697. Crossref, Web of Science, Google Scholar
13. K. Kawamura , On relationship between seven types of Roseman moves, Topology Appl. 196 (2015) 551–557. Crossref, Google Scholar
14. A. Kawauchi, T. Shibuya and S. Suzuki , Descriptions on surfaces in four-space, I; Normal forms, Math. Sem. Notes Kobe Univ. 10 (1982) 72–125. Google Scholar
15. C. Kearton and V. Kurlin , All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron, Algebr. Geom. Topol. 8(3) (2008) 1223–1247. Crossref, Web of Science, Google Scholar
16. S. J. Lomonaco Jr. , 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
17. J. M. Montesinos , A note on twist spun knots, Proc. Amer. Math. Soc. 98 (1986) 180–184. Crossref, Web of Science, Google Scholar
18. M. Polyak , Minimal generating sets of Reidemeister moves, Quantum Topol. 1(4) (2010) 399–411. Crossref, Google Scholar
19. 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
20. F. J. Swenton, Kirby calculator, version 0.931a (2019), available at http://community.middlebury.edu/ $\sim$ mathanimations/klo/. Google Scholar
21. K. Yoshikawa , An enumeration of surfaces in four-space, Osaka J. Math. 31 (1994) 497–522. Web of Science, Google Scholar
22. E. C. Zeeman , Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 417–495. Crossref, Web of Science, Google Scholar