On the 3-loop polynomial of genus 1 knots with trivial Alexander polynomial

˚rhus integral of

[1] D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston, The A

rational homology 3-spheres I: A highly non trivial flat connection on S 3 , Selecta

Math. (N.S.) 8 (2002) 315–339.

[2] D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston, The ˚

Arhus integral of

rational homology 3-spheres II: Invariance and universality, Selecta Math. (N.S.) 8

(2002) 341–371.

[3] D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston, The ˚

Arhus integral of

rational homology 3-spheres III: Relation with the Le-Murakami-Ohtsuki invariant,

Selecta Math. (N.S.) 10 (2004) 305–324.

[4] 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) 1–31.

[5] D. Bar-Natan, R. Lawrence, A rational surgery formula for the LMO invariant, Israel

J. Math. 140 (2004) 29–60.

[6] O. T. Dasbach, On the Combinatorial Structure of Primitive Vassiliev Invariants, II,

J. Comb. Theory, Series A 81 (1998) 127–139.

[7] S. Garoufalidis, A. Kricker, A rational noncommutative invariant of boundary links,

Algebr. Geom. Topol. 8 (2004) 115–204.

[8] S Garoufalidis, A Kricker, A surgery view of boundary links, Math. Ann. 327 (2003)

103115.

[9] S. Garoufalidis, On knots with trivial Alexander polynomial, J. Differential Geometry

67 (2004) 165-191.

[10] S. Garoufalidis, Whitehead doubling persisits, Algebr. Geom. Topol. 4 (2004) 935–942.

60

[11] A. Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture,

arXiv:math. GT/0005284.

[12] J. March´e , On Kontsevich integral of torus knots, Topology Appl. 143 (2004) 15–26.

[13] J. March´e, A computation of the Kontsevich integral of torus knots, Algebr. Geom.

Topol. 4 (2004) 1155–1175.

[14] T. Ohtsuki, Musubime no Fuhenryo (in Japanese), Kyoritsu Shuppan Co.,Ltd 2015.

[15] T. Ohtsuki, Quantum invariants, Series on Knots and Everything 29, World Scientific

Publishing Co., River Edge, NJ (2002).

[16] T. Ohtsuki, A cabling formula for the 2-loop polynomial of knots, Publ. Res. Inst.

Math. Sci. 40 (2004) 949–971.

[17] T. Ohtsuki, On the 2-loop polynomial of knots, Geom. Topol. 11 (2007) 1357–1457.

[18] T. Ohtsuki, D. Moskovich, Vanishing of 3-loop Jacobi diagrams of odd degree, J.

Combin. Theory Ser. A 114 (2007) 919–930.

[19] L. Rozansky, A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial, from: “Invariants of ThreeManifolds (Calgay,1999)”, Topology Appl. 127 (2003) 47–76.

[20] S. Willerton, An almost-integral universal Vassiliev invariant of knots, Algebr. Geom.

Topol. 2 (2002) 649–664

[21] K. Yamaguchi, The 3-loop polynomial of knots obtained by plumbingthe doubles of

two knots,, Internat. J. Math. doi:10.1142/S0129167X23500866

Research Institute for Mathematical Science, Kyoto University, Sakyo-ku, Kyoto, 6068502, Japan

E-mail address: yamakou@kurims.kyoto-u.ac.jp

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