Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T20:35:23.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Burau Maps and Twisted Alexander Polynomials

Part of: Low-dimensional topology

Published online by Cambridge University Press:  27 February 2018

Anthony Conway
Anthony Conway*
Affiliation:
Section de Mathématiques, Université de Genève, 2–4 rue du Lièvre, 1227 Acacias, Geneva, Switzerland ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

Keywords

knotlinkbraid groupBurau representationtwisted Alexander polynomial

MSC classification

Primary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 479 - 497
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abdulrahim, M. N., A faithfulness criterion for the Gassner representation of the pure braid group, Proc. Amer. Math. Soc. 125(5) (1997), 1249–1257.CrossRefGoogle Scholar
2.Bigelow, S., The Burau representation is not faithful for n = 5, Geom. Topol. 3 (1999), 397–404.Google Scholar
3.Birman, J. S., Braids, links, and mapping class groups. Annals of Mathematics Studies, Volume 82 (Princeton University Press, Princeton, NJ, 1975).CrossRefGoogle Scholar
4.Boden, H. U., Dies, E., Gaudreau, A. I., Gerlings, A., Harper, E. and Nicas, A. J., Alexander invariants for virtual knots, J. Knot Theory Ramifications 24(3) (2015), 1550009.Google Scholar
5.Burau, W., Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Univ. Hamburg 11(1) (1935), 179–186.CrossRefGoogle Scholar
6.Cha, J. C. and Friedl, S., Twisted torsion invariants and link concordance, Forum Math. 25(3) (2013), 471–504.Google Scholar
7.Chapman, T. A., Topological invariance of Whitehead torsion, Amer. J. Math. 96 (1974), 488–497.Google Scholar
8.Cimasoni, D., A geometric construction of the Conway potential function, Comment. Math. Helv. 79(1) (2004), 124–146.CrossRefGoogle Scholar
9.Cimasoni, D. and Conway, A., Colored tangles and signatures (arXiv:1507.07818, 2015).Google Scholar
10.Cimasoni, D. and Florens, V., Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360(3) (2008), 1223–1264.Google Scholar
11.Cimasoni, D. and Turaev, V., A Lagrangian representation of tangles, Topology 44(4) (2005), 747–767.Google Scholar
12.Davis, J. F. and Kirk, P., Lecture notes in algebraic topology, Graduate Studies in Mathematics, Volume 35 (American Mathematical Society, Providence, RI, 2001).Google Scholar
13.Fox, R. H., Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560.Google Scholar
14.Friedl, S. and Kim, T., The Thurston norm, fibered manifolds and twisted Alexander polynomials, Topology 45(6) (2006), 929–953.Google Scholar
15.Friedl, S. and Kim, T., Twisted Alexander norms give lower bounds on the Thurston norm, Trans. Amer. Math. Soc. 360(9) (2008), 4597–4618.Google Scholar
16.Friedl, S., Kim, T. and Kitayama, T., Poincaré duality and degrees of twisted Alexander polynomials, Indiana Univ. Math. J. 61(1) (2012), 147–192.Google Scholar
17.Friedl, S. and Vidussi, S., A survey of twisted Alexander polynomials, In The mathematics of knots, Contributions in Mathematical and Computational Sciences, Volume 1, pp. 45–94 (Springer, Heidelberg, 2011).Google Scholar
18.Hillman, J. A., Livingston, C. and Naik, S., Twisted Alexander polynomials of periodic knots, Algebr. Geom. Topol. 6 (2006), 145–169 (electronic).CrossRefGoogle Scholar
19.Jiang, B. J. and Wang, S. C., Twisted topological invariants associated with representations, In Topics in knot theory (Erzurum, 1992), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Volume 399, pp. 211–227 (Kluwer, Dordrecht, 1993).Google Scholar
20.Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38(3) (1999), 635–661.Google Scholar
21.Kirk, P. and Livingston, C., Twisted knot polynomials: inversion, mutation and concordance, Topology 38(3) (1999), 663–671.Google Scholar
22.Kitano, T., Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174(2) (1996), 431–442.Google Scholar
23.Levine, J., Link invariants via the eta invariant, Comment. Math. Helv. 69(1) (1994), 82–119.Google Scholar
24.Lin, X. S., Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17(3) (2001), 361–380.Google Scholar
25.Long, D. D. and Paton, M., The Burau representation is not faithful for n ⩾ 6, Topology 32(2) (1993), 439–447.CrossRefGoogle Scholar
26.Milnor, J., Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426.Google Scholar
27.Morifuji, T., A Torres condition for twisted Alexander polynomials, Publ. Res. Inst. Math. Sci. 43(1) (2007), 143–153.Google Scholar
28.Morton, H., The multivariable Alexander polynomial for a closed braid, In Low-dimensional topology (Funchal, 1998), Contemporary Mathematics, Volume 233, pp. 167–172 (American Mathematical Society, Providence, RI, 1999).Google Scholar
29.Squier, C. C., The Burau representation is unitary, Proc. Amer. Math. Soc. 90(2) (1984), 199–202.Google Scholar
30.Swan, R. G., Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978), 111–120.CrossRefGoogle Scholar
31.Turaev, V., Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41(1(247)) (1986), 97–147, 240.Google Scholar
32.Turaev, V., Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2001).Google Scholar
33.Turaev, V., Faithful linear representations of the braid groups, Astérisque 1999/2000(276) (2002), 389–409. Séminaire Bourbaki.Google Scholar
34.Wada, M., Twisted Alexander polynomial for finitely presentable groups, Topology 33(2) (1994), 241–256.Google Scholar
35.Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38 (Cambridge University Press, Cambridge, 1994).Google Scholar