A note on Riley polynomials of 2-bridge knots
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 5, pp. 1211-1217.

Dans cette note nous montrons l’éxistence d’un epimorphism entre les groupes des noeuds à deux ponts par un argument élémentaire en utilisant le polynôme de Riley. Comme corollaire, nous donnons une classification des noeuds à deux ponts par polynômes de Riley.

In this short note we show the existence of an epimorphism between groups of 2-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of 2-bridge knots by Riley polynomials.

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DOI : 10.5802/afst.1565
Classification : 57M25
Mots clés : Riley polynomial, $2$-bridge knot, epimorphism
Kitano, Teruaki 1 ; Morifuji, Takayuki 2

1 Department of Information Systems Science, Faculty of Science and Engineering, Soka University, Tangi-cho 1-236, Hachioji, Tokyo 192-8577, Japan
2 Department of Mathematics, Hiyoshi Campus, Keio University, Yokohama 223-8521, Japan
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Kitano, Teruaki; Morifuji, Takayuki. A note on Riley polynomials of 2-bridge knots. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 5, pp. 1211-1217. doi : 10.5802/afst.1565. http://www.numdam.org/articles/10.5802/afst.1565/

[1] Burde, Gerhard; Zieschang, Heiner; Heusener, Michael Knots, De Gruyter Studies in Mathematics, 5, Walter de Gruyter, 2014, xiii+417 pages | Zbl

[2] Gordon, Cameron McA.; Luecke, John Knots are determined by their complements, J. Am. Math. Soc., Volume 2 (1989) no. 2, pp. 371-415 | DOI | Zbl

[3] Hartley, Richard; Murasugi, Kunio Homology invariants, Can. J. Math., Volume 30 (1978), pp. 655-670 | DOI | Zbl

[4] Hempel, John Residual finiteness for 3-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984) (Annals of Mathematics Studies), Volume 111, Princeton University Press (1987), pp. 379-396 | Zbl

[5] Malʼtsev, Anatoliù i Ivanovich On isomorphic matrix representations of infinite groups, Rec. Math. Moscou, Volume 8 (1940), pp. 405-422 | Zbl

[6] Riley, Robert Parabolic representations of knot groups, I, Proc. Lond. Math. Soc., Volume 24 (1972), pp. 217-242 | DOI | Zbl

[7] Sakuma, Makoto Epimorphisms between 2-bridge knot groups from the view point of Markoff maps, Intelligence of low dimensional topology 2006 (Hiroshima, Japan) (Series on Knots and Everything), Volume 40, World Scientific, 2007, pp. 279-286 | Zbl

[8] Schubert, Horst Über eine numerische Knoteninvariante, Math. Z., Volume 61 (1954), pp. 245-288 | DOI | Zbl

[9] Scott, Peter The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983), pp. 401-487 | DOI | Zbl

[10] Thurston, William P. Three-dimensional geometry and topology. Vol 1., Princeton Mathematical Series, 35, Princeton University Press, 1997, x+311 pages | Zbl

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