On the 2–loop polynomial of knots

Ohtsuki, Tomotada¹

¹ Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

Geometry & topology, Tome 11 (2007) no. 3, pp. 1357-1475.

Résumé

The 2–loop polynomial of a knot is a polynomial characterizing the 2–loop part of the Kontsevich invariant of the knot. An aim of this paper is to give a methodology to calculate the 2–loop polynomial. We introduce Gaussian diagrams to calculate the rational version of the Aarhus integral explicitly, which constructs the 2–loop polynomial, and we develop methodology of calculating Gaussian diagrams showing many basic formulas of them. As a consequence, we obtain an explicit presentation of the 2–loop polynomial for knots of genus 1 in terms of derivatives of the Jones polynomial of the knots.

Corresponding to quantum and related invariants of 3–manifolds, we can formulate equivariant invariants of the infinite cyclic covers of knots complements. Among such equivariant invariants, we can regard the 2–loop polynomial of a knot as an “equivariant Casson invariant” of the infinite cyclic cover of the knot complement. As an aspect of an equivariant Casson invariant, we show that the 2–loop polynomial of a knot is presented by using finite type invariants of degree $\le 3$ of a spine of a Seifert surface of the knot. By calculating this presentation concretely, we show that the degree of the 2–loop polynomial of a knot is bounded by twice the genus of the knot. This estimate of genus is effective, in particular, for knots with trivial Alexander polynomial, such as the Kinoshita–Terasaka knot and the Conway knot.

Détail

DOI : 10.2140/gt.2007.11.1357

Keywords: knot, Kontsevich invariant, 2–loop polynomial, genus

Bibliographie

[1] D Bar-Natan, S Garoufalidis, On the Melvin–Morton–Rozansky conjecture, Invent. Math. 125 (1996) 103

[2] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3–spheres I: A highly non trivial flat connection on $S^3$, Selecta Math. $($N.S.$)$ 8 (2002) 315

[3] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3–spheres II: Invariance and universality, Selecta Math. $($N.S.$)$ 8 (2002) 341

[4] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3–spheres III: Relation with the Le–Murakami–Ohtsuki invariant, Selecta Math. $($N.S.$)$ 10 (2004) 305

[5] D Bar-Natan, R Lawrence, A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29

[6] D Bar-Natan, T T Q Le, D P Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003) 1

[7] J Conant, Gropes and the rational lift of the Kontsevich integral

[8] D Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc. 59 (1986)

[9] S Garoufalidis, Signatures of links and finite type invariants of cyclic branched covers, from: "Tel Aviv Topology Conference: Rothenberg Festschrift (1998)", Contemp. Math. 231, Amer. Math. Soc. (1999) 87

[10] S Garoufalidis, A Kricker, Finite type invariants of cyclic branched covers, Topology 43 (2004) 1247

[11] S Garoufalidis, A Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004) 115

[12] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1

[13] K Habiro, On the quantum $\mathrm{sl}_2$ invariants of knots and integral homology spheres, from: "Invariants of knots and 3–manifolds (Kyoto, 2001)", Geom. Topol. Monogr. 4, Geom. Topol. Publ., Coventry (2002) 55

[14] K Habiro, T T Q Le, in preparation

[15] V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. $($N.S.$)$ 12 (1985) 103

[16] S Kinoshita, H Terasaka, On unions of knots, Osaka Math. J. 9 (1957) 131

[17] S Kojima, M Yamasaki, Some new invariants of links, Invent. Math. 54 (1979) 213

[18] M Kontsevich, Vassiliev's knot invariants, from: "I. M. Gel'fand Seminar", Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 137

[19] A Kricker, A surgery formula for the 2-loop piece of the LMO invariant of a pair, from: "Invariants of knots and 3-manifolds (Kyoto, 2001)", Geom. Topol. Monogr. 4 (2002) 161

[20] A Kricker, Branched cyclic covers and finite type invariants, J. Knot Theory Ramifications 12 (2003) 135

[21] A Kricker, The lines of the Kontsevich integral and Rozansky's rationality conjecture

[22] T T Q Le, An invariant of integral homology 3–spheres which is universal for all finite type invariants, from: "Solitons, geometry, and topology: on the crossroad" (editors V Buchstaber, S Novikov), Amer. Math. Soc. Transl. Ser. 2 179, Amer. Math. Soc. (1997) 75

[23] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of 3–manifolds, Topology 37 (1998) 539

[24] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997)

[25] J Marché, Cablages et intégrale de Kontsevich rationnelle en bas degré, PhD thesis, Université Paris 7 (2004)

[26] J Marché, Surgery on a single clasper and the 2–loop part of the Kontsevich integral

[27] J Marché, An equivariant Casson invariant of knots in homology spheres, preprint (2005)

[28] D Mullins, The generalized Casson invariant for 2–fold branched covers of $S^3$ and the Jones polynomial, Topology 32 (1993) 419

[29] H Murakami, Quantum $\mathrm{SU}(2)$–invariants dominate Casson's $\mathrm{SU}(2)$–invariant, Math. Proc. Cambridge Philos. Soc. 115 (1994) 253

[30] H Murakami, Quantum $\mathrm{SO}(3)$–invariants dominate the $\mathrm{SU}(2)$–invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc. 117 (1995) 237

[31] J Murakami, T Ohtsuki, Topological quantum field theory for the universal quantum invariant, Comm. Math. Phys. 188 (1997) 501

[32] T Ohtsuki, Problems on invariants of knots and 3–manifolds, from: "Invariants of knots and 3–manifolds (Kyoto, 2001)", Geom. Topol. Monogr. 4 (2002) 377

[33] T Ohtsuki, Quantum invariants, Series on Knots and Everything 29, World Scientific Publishing Co. (2002)

[34] T Ohtsuki, Invariants of knots and 3–dimensional manifolds, Sūgaku 55 (2003) 337

[35] T Ohtsuki, A cabling formula for the 2–loop polynomial of knots, Publ. Res. Inst. Math. Sci. 40 (2004) 949

[36] T Ohtsuki, Equivariant quantum invariants of the infinite cyclic covers of knot complements, preprint, RIMS 15-9 (2005)

[37] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311

[38] P Ozsváth, Z Szabó, Knot Floer homology, genus bounds, and mutation, Topology Appl. 141 (2004) 59

[39] L Rozansky, A rational structure of generating functions for Vassiliev invariants

[40] L Rozansky, A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial, from: "Invariants of Three-Manifolds (Calgary, 1999)", Topology Appl. 127 (2003) 47