This article is based on the lectures in the Winter Braids V (Pau, February 2015). We introduce some studies of twisted Alexander polynomials to non-experts through many concrete examples. In this article we follow the definition of the twisted Alexander polynomial by Wada, which can be defined for a finitely presented group with an epimorphism onto a free abelian group. The main tool is FoxÕs free calculus. In the last two sections we discuss some applications on the fiberedness of a knot and the existence of epimorphisms between knot groups.
@article{WBLN_2015__2__A4_0, author = {Teruaki Kitano}, title = {Introduction to twisted {Alexander} polynomials and related topics}, journal = {Winter Braids Lecture Notes}, note = {talk:4}, publisher = {Winter Braids School}, volume = {2}, year = {2015}, doi = {10.5802/wbln.10}, mrnumber = {3705876}, zbl = {1422.57037}, language = {en}, url = {https://wbln.centre-mersenne.org/articles/10.5802/wbln.10/} }
TY - JOUR TI - Introduction to twisted Alexander polynomials and related topics JO - Winter Braids Lecture Notes N1 - talk:4 PY - 2015 DA - 2015/// VL - 2 PB - Winter Braids School UR - https://wbln.centre-mersenne.org/articles/10.5802/wbln.10/ UR - https://www.ams.org/mathscinet-getitem?mr=3705876 UR - https://zbmath.org/?q=an%3A1422.57037 UR - https://doi.org/10.5802/wbln.10 DO - 10.5802/wbln.10 LA - en ID - WBLN_2015__2__A4_0 ER -
Teruaki Kitano. Introduction to twisted Alexander polynomials and related topics. Winter Braids Lecture Notes, Volume 2 (2015), Talk no. 4, 35 p. doi : 10.5802/wbln.10. https://wbln.centre-mersenne.org/articles/10.5802/wbln.10/
[1] I. Agol and N. M. Dunfield, Certifying the Thurston norm via $\mathit{SL}(2,\u2102)$-twisted homology, arXiv:1501.02136
[2] I. Agol and Y. Liu, Presentation length and Simon’s conjecture, J. Amer. Math. Soc. 25 (2012), no. 1, 151–187. | Article | MR: 2833481 | Zbl: 1237.57002
[3] M. Boileau and S. Boyer, On character varieties, sets of discrete characters and non-zero degree maps, Amer. J. Math. 134 (2012), no. 2, 285–347. | Article | MR: 2904999 | Zbl: 1257.57021
[4] M. Boileau, S. Boyer, A. Reid and S. Wang, Simon’s conjecture for two-bridge knots, Comm. Anal. Geom. 18 (2010), 121–143. | Article | MR: 2660460 | Zbl: 1211.57004
[5] G. Burde, H. Zieschang and M. Heusener, Knots, Third, fully revised and extended edition. De Gruyter Studies in Mathematics, 5. De Gruyter, Berlin, 2014. xiv+417 pp. | Zbl: 1283.57002
[6] J. C. Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4187–4200 | Article | Zbl: 1028.57004
[7] J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.edu/ knotinfo.
[8] J. C. Cha and M. Suzuki, Non-meridional epimorphisms of knot groups, arXiv:1502.06039 | Article | MR: 3493417 | Zbl: 1345.57007
[9] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, in the 1970 Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 329–358 Pergamon, Oxford. | Article
[10] R. Crowell and R. H. Fox, Introduction to knot theory, Reprint of the 1963 original. GTM 57. Springer-Verlag, New York-Heidelberg, 1977. | Article
[11] G. de Rham, Introduction aux polynômes d’un nœud, Enseignement Math. (2) 13 1967 187–194 (1968). | Zbl: 0157.54803
[12] N. Dunfield and I. Agol, Certifying the Thurston norm via $\mathit{SL}(2,\u2102)$-twisted homology, arXiv:1501.02136
[13] N. Dunfield, S. Friedl and N. Jackson, Twisted Alexander polynomials of hyperbolic knots, Exp. Math. 21 (2012), no. 4, 329–352. | Article | MR: 3004250 | Zbl: 1266.57008
[14] R. H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57, (1953). 547–560. | Article | MR: 53938 | Zbl: 0050.25602
[15] R. H. Fox and J. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math. 3 (1966), 257–267. | Zbl: 0146.45501
[16] S. Friedl and T. Kim, The Thurston norm, fibered manifolds and twisted Alexander polynomials, Topology 45 (2006), no. 6, 929–953. | Article | MR: 2263219 | Zbl: 1105.57009
[17] S. Friedl and S. Vidussi, Twisted Alexander polynomials detect fibered 3-manifolds, Ann. of Math. (2) 173 (2011), no. 3, 1587–1643. | Article | MR: 2800721 | Zbl: 1231.57020
[18] S. Friedl and S. Vidussi, A survey of twisted Alexander polynomials, in the book of The mathematics of knots, 45–94, Contrib. Math. Comput. Sci., 1, Springer, Heidelberg, 2011. | Article | Zbl: 1223.57012
[19] S. Friedl and S. Vidussi, A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds, J. Eur. Math. Soc. 15 (2013), no. 6, 2027–2041. | Article | MR: 3120739 | Zbl: 1285.57007
[20] H. Goda, T. Kitano and T. Morifuji, Reidemeister torsion, twisted Alexander polynomial and fibered knots, Comment. Math. Helv. 80 (2005), no. 1, 51–61. | Article | MR: 2130565 | Zbl: 1066.57008
[21] C. Gordon and J. Lucke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. | Article | MR: 965210 | Zbl: 0678.57005
[22] J. Hempel, 3-manifold, Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004.
[23] K. Horie, T. Kitano, M. Matsumoto and M. Suzuki, A partial order on the set of prime knots with up to 11 crossings, J. Knot Theory Ramifications, 20, No. 2 (2011), 275–303. Erratum: J. Knot Theory Ramifications, 21 (2012), no. 4, 1292001, 2 pp. | Article | Zbl: 1218.57004
[24] B. Jiang and S. Wang, Twisted topological invariants associated with representations, in Topics in knot theory (Erzurum, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Acad. Publ., Dordrecht (1993). | Article | Zbl: 0815.55001
[25] D. Johnson, A geometric form of Casson’s invariant and its connection to Reidemeister torsion, unpublished lecture notes.
[26] A. Kawauchi, An imitation theory of manifolds, Osaka J. Math. 26 (1989), no. 3, 447–464. | Zbl: 0701.57014
[27] W. H. Kazez (ed.), Geometric topology, Proceedings of the 1993 Georgia International Topology Conference held at the University of Georgia, Athens, GA, August 2-13, 1993. AMS/IP Studies in Advanced Mathematics, 2.2. American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 1997. xiv+473 pp. | Article
[28] T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), no. 2, 431–442. | Article | MR: 1405595 | Zbl: 0863.57001
[29] T. Kitano and T. Morifuji, Divisibility of twisted Alexander polynomials and fibered knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 1, 179–186. | Zbl: 1117.57004
[30] T. Kitano and T. Morifuji, Twisted Alexander polynomials for irreducible $\mathit{SL}(2,\u2102)$-representations of torus knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 395–406. | Zbl: 1255.57014
[31] T. Kitano and M. Suzuki, A partial order in the knot table, Experiment. Math. 14 (2005), no. 4, 385–390. Erratum: Experiment. Math. 20 (2011), no. 3, 371. | Article | MR: 2193401 | Zbl: 1089.57006
[32] T. Kitano and M. Suzuki, Twisted Alexander polynomials and a partial order on the set of prime knots, in Groups, homotopy and configuration spaces, 307–321, Geom. Topol. Monogr., 13, Geom. Topol. Publ., Coventry, 2008. | Article | Zbl: 1137.57006
[33] T. Kitano and M. Suzuki, Some minimal elements for a partial order of prime knots, arXiv:1412.3168
[34] T. Kitano, M. Suzuki and M. Wada, Twisted Alexander polynomials and surjectivity of a group homomorphism, Algebr. Geom. Topol. 5 (2005), 1315–1324. Erratum: Algebr. Geom. Topol. 11 (2011), 2937–2939 | Article | MR: 2171811 | Zbl: 1081.57004
[35] P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38, (1999), no. 3, 635–661. | Article | MR: 1670420 | Zbl: 0928.57005
[36] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Mathematica Sinica, English Series, 17 (2001), No.3, pp. 361–380 | Article | MR: 1852950 | Zbl: 0986.57003
[37] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations. Reprint of the 1976 second edition. Dover Publications, Inc., Mineola, NY, 2004. | Article | MR: 1534367 | Zbl: 1130.20307
[38] J. Milnor, Two complexes which are homeomorphic but combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590. | Article | MR: 133127 | Zbl: 0102.38103
[39] J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147. | Article | MR: 141115 | Zbl: 0108.36502
[40] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. | Article | MR: 196736 | Zbl: 0147.23104
[41] J. Milnor, Infinite cyclic coverings, 1968 Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), 115–133 Prindle, Weber & Schmidt, Boston, Mass.
[42] J. W. Morgan and H. Bass (ed.), The Smith conjecture, Pure and Applied Mathematics, 112. Academic Press, 1984. xv+243 pp. | Zbl: 0599.57001
[43] T. Morifuji, On a conjecture of Dunfield, Friedl and Jackson, C. R. Math. Acad. Sci. Paris 350 (2012), no. 19-20, 921–924. | Article | MR: 2990904
[44] T. Morifuji, Representation of knot groups into $\mathit{SL}(2;\u2102)$ and twisted Alexander polynomials, Handbook of Group Actions (Vol I) (2015), p527–572, Higher Educational Press and International Press, Beijing-Boston.
[45] T. Morifuji and A. T. Tran, Twisted Alexander polynomials of 2-bridge knots for parabolic representations, Pacific J. Math. 269 (2014), no. 2, 433–451. | Article | MR: 3238485 | Zbl: 1305.57027
[46] L. Neuwirth, On Stallings fibrations, Proc. Amer. Math. Soc. 14 1963 380–381. | Article | MR: 149470 | Zbl: 0117.41002
[47] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between 2-bridge link groups, in the Zieschang Gedenkschrift, Geom. and Topol. Monogr. 14 (2008), 417–450. | Article | Zbl: 1146.57011
[48] D. Rolfsen, Knots and links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990. xiv+439 pp. | Article
[49] R. Riley, Nonabelian representations of 2-bridge knot groups, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 138, 191–208. | Article | MR: 745421 | Zbl: 0549.57005
[50] H. Seifert, Über das Geschlecht von Knoten, (German) Math. Ann. 110 (1935), no. 1, 571–592. | Article | Zbl: 0010.13303
[51] D.S. Silver and W. Whitten, Knot group epimorphisms, J. Knot Theory Ramifications 15 (2006), 153–166. | Article | MR: 2207903 | Zbl: 1096.57007
[52] D.S. Silver and W. Whitten, Knot group epimorphisms II, preprint.
[53] D. Silver and S. G. Wiliams, Twisted Alexander polynomials detect the unknot, Algebr. Geom. Topol. 6 (2006), 1893–1901. | Article | MR: 2263053 | Zbl: 1132.57010
[54] J. Stallings, On fibering certain 3-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), (1962) pp. 95–100 Prentice-Hall, Englewood Cliffs, N.J.
[55] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), 241–256. | Article | MR: 1273784 | Zbl: 0822.57006
[56] C. T. C. Wall, Surgery on compact manifolds, Second edition. Edited and with a foreword by A. A. Ranicki. Mathematical Surveys and Monographs, 69. AMS, Providence, RI, 1999. | Article
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