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2017 | OriginalPaper | Buchkapitel

1. Link Invariants from the Yokonuma–Hecke Algebras

verfasst von : Konstantinos Karvounis, Sofia Lambropoulou

Erschienen in: Algebraic Modeling of Topological and Computational Structures and Applications

Verlag: Springer International Publishing

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Abstract

The Yokonuma–Hecke algebras are naturally related to the framed braid group and they support a Markov trace. Consequently, invariants for various types of links (framed, classical, singular and transverse) are derived from these algebras. In this paper, we present results about these invariants and their properties. We focus, in particular, on the family of 2-variable classical link invariants that are not topologically equivalent to the HOMFLYPT polynomial and on the 3-variable classical link invariant that generalizes this family and the HOMFLYPT polynomial.

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Nächstes Kapitel A Survey on Temperley–Lieb-type Quotients from the Yokonuma–Hecke Algebras

Private communication with F. Aicardi

Aicardi, F., Juyumaya, J.: Markov trace on the algebra of braids and ties. Mosc. Math. J. 16(3), 397–431 (2016)MathSciNetMATH

Aicardi, F., Juyumaya, J.: Tied links. J. Knot Theory Ramif. 25(9), 1641001 (2016)CrossRef

Aicardi, F., Juyumaya, J.: Kauffman type invariants for tied links. arXiv:1607.04841 [math.GT]

Baez, J.C.: Link invariants of finite type and perturbation theory. Lett. Math. Phys. 26(1), 43–51 (1992)MathSciNetCrossRefMATH

Bennequin, D.: Entrelacements et équations de Pfaffe. Asterisque 107–108, 87–161 (1983)MATH

Birman, J.S.: New points of view in knot theory. Bull. Am. Math. Soc. (N.S.) 28(2), 253–287 (1993)

Chmutov, S., Duzhin, S., Mostovoy, Y.: Introduction to Vassiliev Knot Invariants. Cambridge University Press, Cambridge (2012)CrossRefMATH

Cha, J.C., Livingston, C.: LinkInfo: Table of Knot Invariants. http://www.indiana.edu/~linkinfo (2015). Accessed 16 April 2015

10.

Chmutov, S., Jablan, S., Karvounis, K., Lambropoulou, S.: On the link invariants from the Yokonuma-Hecke algebras. J. Knot Theory Ramif. 25(9), 1641004 (2016)MathSciNetCrossRefMATH

11.

Chlouveraki, M., Juyumaya, J., Karvounis, K., Lambropoulou, S.: Identifying the invariants for classical knots and links from the Yokonuma–Hecke algebra, submitted for publication. arXiv: 1505.06666 [math.GT]

12.

Chlouveraki, M., Lambropoulou, S.: The Yokonuma-Hecke algebras and the HOMFLYPT polynomial. J. Knot Theory Ramif. 22(14), 1350080 (2013)MathSciNetCrossRefMATH

13.

Chlouveraki, M., Poulain d’Andecy, L.: Representation theory of the Yokonuma-Hecke algebra. Adv. Math. 259, 134–172 (2014)

14.

Chlouveraki, M., Poulain d’Andecy, L.: Markov traces on affine and cyclotomic Yokonuma-Hecke algebras. Int. Math. Res. Not. 2016(14), 4167–4228 (2016)

15.

Chlouveraki, M., Pouchin, G.: Determination of the representations and a basis for the Yokonuma-Temperley-Lieb algebra. Algebras Represent. Theory 18(2), 421–447 (2015)MathSciNetCrossRefMATH

16.

Chlouveraki, M., Pouchin, G.: Representation theory and an isomorphism theorem for the Framisation of the Temperley–Lieb algebra. Math. Z., 11–24 (2016)

17.

Espinoza, J., Ryom-Hansen, S.: Cell structures for the Yokonuma–Hecke algebra and the algebra of braids and ties, submitted for publication. arXiv:1506.00715

18.

Flores, M., Juyumaya, J., Lambropoulou, S.: A Framization of the Hecke algebra of Type B. arXiv:1603.08487

19.

Freyd, P., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A., Yetter, D.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12(2), 239–246 (1985)

20.

Fuchs, D., Tabachnikov, S.: Invariants of Legendrian and transverse knots in the standard contact space. Topology 36(5), 1025–1053 (1997)MathSciNetCrossRefMATH

21.

Gemein, B.: Singular braids and Markov’s theorem. J. Knot Theory Ramif. 6(4), 441–454 (1997)MathSciNetCrossRefMATH

22.

Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: The Yokonuma-Temperley-Lieb algebras. Banach Cent. Pub. 103, 73–95 (2014)

23.

Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: Framization of the Temperley–Lieb algebra, to appear in Mathematical Research Letters. arXiv:1304.7440

24.

Goundaroulis, D., Lambropoulou, S.: Classical link invariants from the framization of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A, to appear in J. Knot Theory Ramif. arXiv:1608.01812v1

25.

Goundaroulis, D., Lambropoulou, S.: A new two-variable generalization of the Jones polynomial. arXiv:1608.01812v1

26.

Jacon, N.: Poulain d’Andecy, L.: An isomorphism theorem for Yokonuma-Hecke algebras and applications to link invariants. Math. Z. 283(1–2), 301–338 (2016)MathSciNetCrossRefMATH

27.

Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126(2), 335–388 (1987)MathSciNetCrossRefMATH

28.

Juyumaya, J.: Another algebra from the Yokonuma–Hecke algebra. ICTP Preprint IC/1999/160

29.

Juyumaya, J.: Markov trace on the Yokonuma-Hecke algebra. J. Knot Theory Ramif. 13, 25–39 (2004)MathSciNetCrossRefMATH

30.

Juyumaya, J., Lambropoulou, S.: \(p\)-adic framed braids. Topol. Appl. 154, 149–191 (2013)MathSciNetMATH

31.

Juyumaya, J., Lambropoulou, S.: \(p\)-adic framed braids II. Adv. Math. 234, 149–191 (2013)MathSciNetCrossRefMATH

32.

Juyumaya, J., Lambropoulou, S.: An adelic extension of the Jones polynomial. In: Banagl, M., Vogel, D. (eds.) The Mathematics of Knots, Contributions in the Mathematical and Computational Sciences, Vol. 1, Springer, Berlin. arXiv:0909.2545v2 [math.GT]

33.

Juyumaya, J., Lambropoulou, S.: An invariant for singular knots. J. Knot Theory Ramif. 18(6), 825–840 (2009)MathSciNetCrossRefMATH

34.

Juyumaya, J., Lambropoulou, S.: On the framization of knot algebras, in New Ideas in Low-Dimensional Topology. In: Kaufffman, L.H., Manturov, V. (eds.) Series Knots Everything. World Scientific Press, Singapore (2014)

35.

Karvounis, K.: Enabling computations for link invariants coming from the Yokonuma-Hecke algebras. J. Knot Theory Ramif. 25(9), 1641012 (2016)MathSciNetCrossRefMATH

36.

Kauffman, L.H., Lambropoulou, S.: New invariants of links and their state sum models. arXiv:1703.03655 [math.GT]

37.

Ko, K.H., Smolinsky, L.: The framed braid group and \(3\)-manifolds. Proc. AMS 115(2), 541–551 (1992)MathSciNetMATH

38.

Lambropoulou, S.: L-moves and Markov theorems. J. Knot Theory Ramif. 16(10), 1–10 (2007)MathSciNetCrossRefMATH

39.

Lickorish, W.B.R., Millett, K.: A polynomial invariant of oriented links. Topology 26(1), 107–141 (1987)MathSciNetCrossRefMATH

40.

Lusztig, G.: Character sheaves on disconnected groups VII. Represent. Theory 9, 209266 (2005)MathSciNetCrossRefMATH

41.

Orevkov, S.Y., Shevchishin, V.V.: Markov theorem for transversal links, J. Knot Theory Ramif. 12(7) (2003) 905–913. Preprint arXiv:math/0112207v2 [math.GT]

42.

Poulain d’Andecy, L., Wagner, E.: The HOMFLYPT polynomials of sublinks and the Yokonuma–Hecke algebras. Preprint arXiv:1606.00237v1 [math.GT]

43.

Przytycki, J.H., Traczyk, P.: Invariants of links of Conway type. Kobe J. Math. 4(2), 115–139 (1988)MathSciNetMATH

44.

Smolin, L.: Knot theory, loop space and the diffeomorphism group. New perspectives in canonical gravity, Monographs Textbooks Physics Science Lecture Notes, Vol. 5, pp. 245–266. Bibliopolis, Naples (1988)

45.

Thiem, N.: Unipotent Hecke algebras of \({\text{ GL }}_n(\mathbb{F}_q)\). J. Algebra 284, 559–577 (2005)MathSciNetCrossRefMATH

46.

Wrinkle, N.: The Markov Theorem for Transverse Knots. Ph.D. thesis, Columbia University (2002). arXiv:math/0202055v1 [math.GT]

47.

Yokonuma, T.: Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini. C.R. Acad. Sci. Paris 264, 344–347 (1967)

Titel: Link Invariants from the Yokonuma–Hecke Algebras
verfasst von: Konstantinos Karvounis
Sofia Lambropoulou
Verlag: Springer International Publishing
Buch: Algebraic Modeling of Topological and Computational Structures and Applications
Print ISBN: 978-3-319-68102-3

Electronic ISBN: 978-3-319-68103-0

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-68103-0_1

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