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http://dx.doi.org/10.4134/BKMS.b170258

KAUFFMAN POLYNOMIAL OF PERIODIC KNOTTED TRIVALENT GRAPHS  

Aboufattoum, Ayman (Department of Mathematics Faculty of Science Kuwait University)
Elsakhawy, Elsyed A. (Department of Mathematics Faculty of Science Ain Shams University)
Istvan, Kyle (Department of Mathematics Louisiana State University)
Qazaqzeh, Khaled (Department of Mathematics Faculty of Science Kuwait University)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.3, 2018 , pp. 799-808 More about this Journal
Abstract
We generalize some of the congruences in [20] to periodic knotted trivalent graphs. As an application, a criterion derived from one of these congruences is used to obstruct periodicity of links of few crossings for the odd primes p = 3, 5, 7, and 11. Moreover, we derive a new criterion of periodic links. In particular, we give a sufficient condition for the period to divide the crossing number. This gives some progress toward solving the well-known conjecture that the period divides the crossing number in the case of alternating links.
Keywords
Kauffman polynomial; periodic links; knotted trivalent graphs; crossing number; adequate links;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 S. Jabuka and S. Naik, Periodic knots and Heegaard Floer correction terms, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1651-1674.   DOI
2 L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417-471.   DOI
3 W. B. R. Lickorish, Some link-polynomial relations, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 1, 103-107.   DOI
4 W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer-Verlag, New York, 1997.
5 K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174.   DOI
6 K. Murasugi, On symmetry of knots, Tsukuba J. Math. 4 (1980), no. 2, 331-347.   DOI
7 K. Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988), no. 2, 319-329.   DOI
8 S. Naik, Periodicity, genera and Alexander polynomials of knots, Pacific J. Math. 166 (1994), no. 2, 357-371.   DOI
9 S. Naik, New invariants of periodic knots, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 2, 281-290.   DOI
10 J. H. Przytycki, On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283 (1989), no. 3, 465-478.   DOI
11 M. B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93 (1988), no. 2, 285-296.   DOI
12 P. Traczyk, 10101 has no period 7: a criterion for periodic links, Proc. Amer. Math. Soc. 108 (1990), no. 3, 845-846.   DOI
13 P. Traczyk, Periodic knots and the skein polynomial, Invent. Math. 106 (1991), no. 1, 73-84.   DOI
14 Y. Yokota, The Jones polynomial of periodic knots, Proc. Amer. Math. Soc. 113 (1991), no. 3, 889-894.   DOI
15 Y. Yokota, The Kauffman polynomial of periodic knots, Topology 32 (1993), no. 2, 309-324.   DOI
16 C. Adams, M. Hildebrand, and J.Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc. 326 (1991), no. 1, 1-56.   DOI
17 D. Bar-Natan and S. Morrison, The Mathematica package KnotTheory. The Knot Atlas, http://katlas.math.toronto.edu/wiki/.
18 G. Burde and H. Zieschang, Knots, second edition, De Gruyter Studies in Mathematics, 5, Walter de Gruyter & Co., Berlin, 2003.
19 C. Caprau and J. Tipton, The Kauffman polynomial and trivalent graphs, Kyungpook Math. J. 55 (2015), no. 4, 779-806.   DOI
20 N. Chbili, Strong periodicity of links and the coefficients of the Conway polynomial, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2217-2224.   DOI
21 N. Chbili, Le polynome de Hom y des noeuds librement periodiques, C. R. Acad. Sci. Paris Ser. I Math. 325 (1997), no. 4, 411-414.   DOI
22 N. Chbili, Equivalent Khovanov homology associated with symmetric links, Kobe J. Math. 27 (2010), no. 1-2, 73-89.
23 J. F. Davis and C. Livingston, Alexander polynomials of periodic knots, Topology 30 (1991), no. 4, 551-564.   DOI
24 K. Hendricks, A note on the Floer homology of doubly-periodic knots. Preprint, arXiv:1206.5989v1.
25 J. A. Hillman, C. Livingston, and S. Naik, Twisted Alexander polynomials of periodic knots, Algebr. Geom. Topol. 6 (2006), 145-169.   DOI