Year 2021,
Volume: 4 Issue: 1, 59 - 66, 01.03.2021
Kemal Taşköprü
,
Zekiye Şevval Sinan
References
- [1] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30 (1928), 275–308.
- [2] J. H. Conway, An enumeration of knots and kinks, and some of their algebraic properties, in Computational Problems in Abstract Algebra, (Pergamon, Oxford, 1970), 329–358.
- [3] S. Friedl, S. Vidussi, A Survey of Twisted Alexander Polynomials, M. Banagl, D. Vogel (eds), The Mathematics of Knots. Contributions in Mathematical and Computational Sciences, Springer, Berlin, 2011, pp. 45–94.
- [4] İ. Altıntaş, K. Taşköprü, A generalization of the Alexander polynomial, Int. J. Math. Comb., 4 (2016), 21–28.
- [5] İ. Altıntaş, K. Taşköprü, A generalization of the Alexander polynomial as an application of the delta derivative, Turk. J. Math., 42(2) (2018), 515–527.
- [6] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc., 12 (1985), 103–111.
- [7] L. H. Kauffman, State models and the Jones polynomial, Topology, 26 (1987), 395–407.
- [8] İ. Altıntaş, An oriented state model for the Jones polynomial and its applications alternating links, Appl. Math. Comput., 194(1) (2007), 168–178.
- [9] P. M. Melvin, H. R. Morton, The coloured Jones function, Commun. Math. Phys, 169 (1995), 501–520.
- [10] İ. Altıntaş, Computer algebra and colored Jones polynomials, Appl. Math. Comput., 182(1) (2006), 804–808.
- [11] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneau, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12 (1985), 239–246.
- [12] J. H. Przytycki, P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc., 100(4) (1987), 744–748.
- [13] H. Doll, J. Hoste, A tabulation of oriented links, Math. Comp., 57(196) (1991), 747–761.
- [14] R. D. Brandt, W. B. R. Lickorish, K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math., 84 (1986), 563–573.
- [15] C. F. Ho, A new polynomial for knots and links–preliminary report, Abstracts Amer. Math. Soc., 6 (1985), 300.
- [16] L. H. Kauffman, On knots, vol. 115 of Annals of Mathematics Study, Princeton University Press, New Jersey, 1987.
- [17] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., 318 (1990), 417–471.
- [18] Y. Allin, M. E. Bozh¨uy¨uk, The group of twist knots, Math. Comput. Appl., 1(2) (1996), 7-15.
- [19] J. Hoste, P. D. Shanahan, Trace fields of twist knots, J. Knot Theory Ramifications, 10(4) (2001), 625-639.
- [20] J. Hoste, P. D. Shanahan, A formula for the A-polynomial of twist knots, J. Knot Theory Ramifications, 13(2) (2004), 193-209.
- [21] J. Dubois, V. Huynh, Y. Yamaguchi, Non-abelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications, 18(3) (2009), 303–341.
- [22] S. Garoufalidis, X. Sun, The non-commutative A-polynomial twist knots, J. Knot Theory Ramifications, 19(12) (2010), 1571-1595.
- [23] D. V. Mathews, An explicit formula for the A-polynomial of twist knots, J. Knot Theory Ramifications, 23(9) (2014), 1450044, 5 pages.
- [24] A. T. Tran, Reidemeister torsion and Dehn surgery on twist knots, Tokyo J. Math., 39(2) (2016), 517-526.
- [25] A. Şahin, B. Şahin, Jones polynomial for graphs of twist knots, Appl. Appl. Math., 14(2) (2019), 1269-1278.
- [26] B. Berceanu, A. R. Nizami, A recurrence relation for the Jones polynomial, J. Korean Math. Soc., 51 (2014), 443–462.
- [27] P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, J. Symbolic Comput., 68 (2015), 215–229.
- [28] S. Duzhin, M. Shkolnikov, A formula for the HOMFLY polynomial of rational links, Arnold Math. J., 1 (2015), 345–359.
- [29] K. Taşköprü, İ. Altıntaş, HOMFLY polynomials of torus links as generalized Fibonacci polynomials, Electron. J. Combin., 22 (2015), 4.8.
- [30] İ. Altıntaş, K. Taşköprü, M. Beyaztaş, Bracket polynomials of torus links as Fibonacci polynomials, Int. J. Adv. Appl. Math. and Mech., 5(3) (2018), 35–43.
- [31] İ. Altıntas¸, K. Taşköprü, Unoriented knot polynomials of torus links as Fibonacci-type polynomials, Asian-Eur. J. Math., 12(1) (2019), 1950053, 17 pages.
Recurrence Relations for Knot Polynomials of Twist Knots
Year 2021,
Volume: 4 Issue: 1, 59 - 66, 01.03.2021
Kemal Taşköprü
,
Zekiye Şevval Sinan
Abstract
This paper gives HOMFLY polynomials and Kauffman polynomials L and F of twist knots as recurrence relations, respectively, and also provides some recursive properties of them.
References
- [1] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc., 30 (1928), 275–308.
- [2] J. H. Conway, An enumeration of knots and kinks, and some of their algebraic properties, in Computational Problems in Abstract Algebra, (Pergamon, Oxford, 1970), 329–358.
- [3] S. Friedl, S. Vidussi, A Survey of Twisted Alexander Polynomials, M. Banagl, D. Vogel (eds), The Mathematics of Knots. Contributions in Mathematical and Computational Sciences, Springer, Berlin, 2011, pp. 45–94.
- [4] İ. Altıntaş, K. Taşköprü, A generalization of the Alexander polynomial, Int. J. Math. Comb., 4 (2016), 21–28.
- [5] İ. Altıntaş, K. Taşköprü, A generalization of the Alexander polynomial as an application of the delta derivative, Turk. J. Math., 42(2) (2018), 515–527.
- [6] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc., 12 (1985), 103–111.
- [7] L. H. Kauffman, State models and the Jones polynomial, Topology, 26 (1987), 395–407.
- [8] İ. Altıntaş, An oriented state model for the Jones polynomial and its applications alternating links, Appl. Math. Comput., 194(1) (2007), 168–178.
- [9] P. M. Melvin, H. R. Morton, The coloured Jones function, Commun. Math. Phys, 169 (1995), 501–520.
- [10] İ. Altıntaş, Computer algebra and colored Jones polynomials, Appl. Math. Comput., 182(1) (2006), 804–808.
- [11] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneau, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc., 12 (1985), 239–246.
- [12] J. H. Przytycki, P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc., 100(4) (1987), 744–748.
- [13] H. Doll, J. Hoste, A tabulation of oriented links, Math. Comp., 57(196) (1991), 747–761.
- [14] R. D. Brandt, W. B. R. Lickorish, K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math., 84 (1986), 563–573.
- [15] C. F. Ho, A new polynomial for knots and links–preliminary report, Abstracts Amer. Math. Soc., 6 (1985), 300.
- [16] L. H. Kauffman, On knots, vol. 115 of Annals of Mathematics Study, Princeton University Press, New Jersey, 1987.
- [17] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., 318 (1990), 417–471.
- [18] Y. Allin, M. E. Bozh¨uy¨uk, The group of twist knots, Math. Comput. Appl., 1(2) (1996), 7-15.
- [19] J. Hoste, P. D. Shanahan, Trace fields of twist knots, J. Knot Theory Ramifications, 10(4) (2001), 625-639.
- [20] J. Hoste, P. D. Shanahan, A formula for the A-polynomial of twist knots, J. Knot Theory Ramifications, 13(2) (2004), 193-209.
- [21] J. Dubois, V. Huynh, Y. Yamaguchi, Non-abelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications, 18(3) (2009), 303–341.
- [22] S. Garoufalidis, X. Sun, The non-commutative A-polynomial twist knots, J. Knot Theory Ramifications, 19(12) (2010), 1571-1595.
- [23] D. V. Mathews, An explicit formula for the A-polynomial of twist knots, J. Knot Theory Ramifications, 23(9) (2014), 1450044, 5 pages.
- [24] A. T. Tran, Reidemeister torsion and Dehn surgery on twist knots, Tokyo J. Math., 39(2) (2016), 517-526.
- [25] A. Şahin, B. Şahin, Jones polynomial for graphs of twist knots, Appl. Appl. Math., 14(2) (2019), 1269-1278.
- [26] B. Berceanu, A. R. Nizami, A recurrence relation for the Jones polynomial, J. Korean Math. Soc., 51 (2014), 443–462.
- [27] P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, J. Symbolic Comput., 68 (2015), 215–229.
- [28] S. Duzhin, M. Shkolnikov, A formula for the HOMFLY polynomial of rational links, Arnold Math. J., 1 (2015), 345–359.
- [29] K. Taşköprü, İ. Altıntaş, HOMFLY polynomials of torus links as generalized Fibonacci polynomials, Electron. J. Combin., 22 (2015), 4.8.
- [30] İ. Altıntaş, K. Taşköprü, M. Beyaztaş, Bracket polynomials of torus links as Fibonacci polynomials, Int. J. Adv. Appl. Math. and Mech., 5(3) (2018), 35–43.
- [31] İ. Altıntas¸, K. Taşköprü, Unoriented knot polynomials of torus links as Fibonacci-type polynomials, Asian-Eur. J. Math., 12(1) (2019), 1950053, 17 pages.