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

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