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)
📌 3 Key Takeaways
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Objective & Hypothesis: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).
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Key Findings & Data:Then, by adapting some results on Fox n-colorings to AS(Zn, B), we obtain some new results over this semigroup.
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Conclusion & Impact: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)
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Knot Theory · Semi Group · Quandle
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)
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