SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS

Year 2024, Volume: 7 Issue: 2, 113 - 127, 31.07.2024

Umut Esen Ahmet Sinan Çevik Mehmet Çitil

https://doi.org/10.33773/jum.1504811

Abstract

Abstract. For a coloring set B ⊆ Zn, by considering the Fox n-coloring of any knot K and using the knot semigroup KS, we show that the set B is actually the same with the set C in the alternating sum semigroup AS(Zn, C). Then, by adapting some results on Fox n-colorings to AS(Zn, B), we obtain some new results over this semigroup. In addition, we present the existence of different homomorphisms (or different isomorphisms in some cases) between the semigroups KS and AS(Zn, B), and then obtained the number of homomorphisms is in fact a knot invariant. Moreover, for different knots K1 and K2
, we establish one can obtain a homomorphism or an isomorphism from the different knot semigroups K1S and K2S
to the same alternating sum semigroup AS(Zn, B)

Keywords

Knot Theory, Semi Group, Quandle

References

• C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, (2004).
• J. W. Alexander, Topological invariants of knots and links, Trans. Am. Math. Soc., Vol.30, pp.275-306 (1923).
• P. Andersson, The color invariant for knots and links, Amer. Math. Monthly, Vol.5, No.102, pp.442-448 (1995).
• Y. Bae, Coloring link diagrams by Alexander quandles, Journal of Knot Theory and Its Ramifications, Vol.21, No.10, pp.13 (2012).
• A. L. Breiland, L. Oesper, L. Taalman, p-coloring Classes of Torus Knots, Missouri J. Math. Sci., Vol.21, pp.120-126 (2009).
• K. Brownell, K. O'Neil, L. Taalman, Counting m-coloring classes of knots and links, Pi Mu Epsilon Journal, Vol.12, No.5, pp.265-278 (2005).
• W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, Journal of Knot Theory and its Ramifications, Vol.23, No.6, (2014).
• R. H. Crowell, R. H. Fox, Introduction to Knot Theory, Springer, (1977).
• M. Elhamdadi, S. Nelson, Quandles: An Introduction to the Algebra of Knots, AMS Book Series, Vol.74, (2015).
• D. Joyce, A Classifying Invariant of Knots, The Knot quandle, Journal of Pure and Applied Algebra, Vol.23, pp.37-65 (1982).
• L. H. Kauffman, Formal Knot Theory, Dover Publications, (2006).
• A. Kawauchi, A Survey of Knot Theory, Springer, (1996).
• W. B. R. Lickorish, An introduction to Knot Theory, Springer, (1997).
• T. Mochizuki, The Third Cohomology Groups of Dihedral Quandles, J. Knot Theory Ramifications, Vol.20, No.7, pp.1041-1057 (2011).
• K. Murasugi, Classical Knot Invariants and Elementary Number Theory, Contemporary Mathematics, pp.167-196, (2006).
• O. Nanyes, An Elementary Proof that the Borromean Rings are Non-Splittable, The American Mathematical Monthly, Vol.8, No.100, pp.786-789 (1993).
• D. Rolfsen, Knots and links, AMS Chelsea Pub, (2003).
• A. Vernitski, Describing semigroups with defining relations of the form xy = yz and yx = zy and connections with knot theory, Semigroup Forum, Vol.93, No.2, pp.387-402 (2016).
• N. Yoko, On the Alexander polynomials of pretzel links L, Kobe J. Math, Vol.2, pp.167-177 (1987).