Twisted Alexander polynomials of periodic knots

Hillman, Jonathan A¹ ; Livingston, Charles² ; Naik, Swatee³

¹School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia
²Department of Mathematics, Indiana University, Bloomington IN 47405, USA
³Department of Mathematics and Statistics, University of Nevada, Reno NV 89557, USA

Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 145-169

Résumé

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings.

Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley’s criterion does not apply.

Détail

DOI : 10.2140/agt.2006.6.145

Keywords: twisted alexander polynomial, periodic knot

Bibliographie

[1] C Adams, M Hildebrand, J Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc. 326 (1991) 1

[2] M Boileau, B Zimmermann, Symmetries of nonelliptic Montesinos links, Math. Ann. 277 (1987) 563

[3] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)

[4] G Burde, Über periodische Knoten, Arch. Math. $($Basel$)$ 30 (1978) 487

[5] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter Co. (2003)

[6] J C Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003) 4187

[7] N Chbili, The skein polynomial of freely periodic knots, Topology Appl. 121 (2002) 535

[8] J I Cogolludo, V Florens, Twisted Alexander polynomials of plane algebraic curves,

[9] J F Davis, C Livingston, Alexander polynomials of periodic knots, Topology 30 (1991) 551

[10] J F Davis, C Livingston, Periodic knots, Smith theory, and Murasugi's congruence, Enseign. Math. $(2)$ 37 (1991) 1

[11] C H Dowker, M B Thistlethwaite, Classification of knot projections, Topology Appl. 16 (1983) 19

[12] A L Edmonds, Least area Seifert surfaces and periodic knots, Topology Appl. 18 (1984) 109

[13] S Friedl, T Kim, The Thurston norm, fibered manifolds and twisted Alexander polynomials,

[14] H Goda, T Kitano, T Morifuji, Reidemeister torsion, twisted Alexander polynomial and fibered knots, Comment. Math. Helv. 80 (2005) 51

[15] H Goda, T Morifuji, Twisted Alexander polynomial for $\mathrm{SL}(2,\mathbb C)$-representations and fibered knots, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003) 97

[16] R Hartley, Knots with free period, Canad. J. Math. 33 (1981) 91

[17] J A Hillman, New proofs of two theorems on periodic knots, Arch. Math. $($Basel$)$ 37 (1981) 457

[18] J A Hillman, On the Alexander polynomial of a cyclically periodic knot, Proc. Amer. Math. Soc. 89 (1983) 155

[19] J Hillman, Algebraic invariants of links, Series on Knots and Everything 32, World Scientific Publishing Co. (2002)

[20] J Hoste, M Thistlethwaite, Knotscape

[21] B J Jiang, S C Wang, Twisted topological invariants associated with representations, from: "Topics in knot theory (Erzurum, 1992)", NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Acad. Publ. (1993) 211

[22] P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635

[23] P Kirk, C Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999) 663

[24] T Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996) 431

[25] T Kitano, M Suzuki, A partial order in the knot table, University of Tokyo Graduate School of Mathematical Sciences, UTMS 2005–7 (March 8, 2005)

[26] K Kodama, M Sakuma, Symmetry groups of prime knots up to 10 crossings, from: "Knots 90 (Osaka, 1990)", de Gruyter (1992) 323

[27] W P Li, L Xu, Counting $\mathrm{SL}_2(\mathbf{F}_{2^s})$ representations of torus knot groups, Acta Math. Sin. $($Engl. Ser.$)$ 19 (2003) 233

[28] X S Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. $($Engl. Ser.$)$ 17 (2001) 361

[29] C Livingston, J C Cha, KnotInfo: Table of Knots

[30] T Morifuji, A Torres condition for twisted Alexander polynomials,

[31] T Morifuji, A twisted invariant for finitely presentable groups, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000) 143

[32] K Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971) 162

[33] S Naik, Periodicity, genera and Alexander polynomials of knots, Pacific J. Math. 166 (1994) 357

[34] S Naik, New invariants of periodic knots, Math. Proc. Cambridge Philos. Soc. 122 (1997) 281

[35] J H Przytycki, On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283 (1989) 465

[36] M Sakuma, On the polynomials of periodic links, Math. Ann. 257 (1981) 487

[37] M Sakuma, Non-free-periodicity of amphicheiral hyperbolic knots, from: "Homotopy theory and related topics (Kyoto, 1984)", Adv. Stud. Pure Math. 9, North-Holland (1987) 189

[38] A Tamulis, Knots of ten or fewer crossings of algebraic order 2, J. Knot Theory Ramifications 11 (2002) 211

[39] P Traczyk, $10_{101}$ has no period 7: a criterion for periodic links, Proc. Amer. Math. Soc. 108 (1990) 845

[40] H F Trotter, Periodic automorphisms of groups and knots, Duke Math. J. 28 (1961) 553

[41] M Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241

[42] J Weeks, SnapPea

[43] Y Yokota, The skein polynomial of periodic knots, Math. Ann. 291 (1991) 281