Araştırma Makalesi
BibTex RIS Kaynak Göster

SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS

Yıl 2024, , 113 - 127, 31.07.2024
https://doi.org/10.33773/jum.1504811

Öz

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)

Kaynakça

  • 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).
Yıl 2024, , 113 - 127, 31.07.2024
https://doi.org/10.33773/jum.1504811

Öz

Kaynakça

  • 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).
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Grup Teorisi ve Genellemeler, Topoloji
Bölüm Araştırma Makalesi
Yazarlar

Umut Esen 0000-0001-9697-8502

Ahmet Sinan Çevik 0000-0002-7539-5065

Mehmet Çitil 0000-0003-3899-3434

Yayımlanma Tarihi 31 Temmuz 2024
Gönderilme Tarihi 25 Haziran 2024
Kabul Tarihi 23 Temmuz 2024
Yayımlandığı Sayı Yıl 2024

Kaynak Göster

APA Esen, U., Çevik, A. S., & Çitil, M. (2024). SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS. Journal of Universal Mathematics, 7(2), 113-127. https://doi.org/10.33773/jum.1504811
AMA Esen U, Çevik AS, Çitil M. SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS. JUM. Temmuz 2024;7(2):113-127. doi:10.33773/jum.1504811
Chicago Esen, Umut, Ahmet Sinan Çevik, ve Mehmet Çitil. “SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS”. Journal of Universal Mathematics 7, sy. 2 (Temmuz 2024): 113-27. https://doi.org/10.33773/jum.1504811.
EndNote Esen U, Çevik AS, Çitil M (01 Temmuz 2024) SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS. Journal of Universal Mathematics 7 2 113–127.
IEEE U. Esen, A. S. Çevik, ve M. Çitil, “SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS”, JUM, c. 7, sy. 2, ss. 113–127, 2024, doi: 10.33773/jum.1504811.
ISNAD Esen, Umut vd. “SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS”. Journal of Universal Mathematics 7/2 (Temmuz 2024), 113-127. https://doi.org/10.33773/jum.1504811.
JAMA Esen U, Çevik AS, Çitil M. SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS. JUM. 2024;7:113–127.
MLA Esen, Umut vd. “SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS”. Journal of Universal Mathematics, c. 7, sy. 2, 2024, ss. 113-27, doi:10.33773/jum.1504811.
Vancouver Esen U, Çevik AS, Çitil M. SOME COLORING RESULTS ON SPECIAL SEMIGROUPS OBTAINED FROM PARTICULAR KNOTS. JUM. 2024;7(2):113-27.