Epimorphisms and Boundary Slopes of 2-Bridge Knots
| Autor: | Hoste, Jim, Shanahan, Patrick D. |
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| Rok vydání: | 2010 |
| Předmět: | |
| Zdroj: | Algebr. Geom. Topol. 10 (2010) 1221-1244 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/agt.2010.10.1221 |
| Popis: | In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be 2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which, for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'} with K_{p'/q'}>K_{p/q}. After characterizing all 2-bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, K_{p'/q'} is either a torus knot or has 5 or more distinct boundary slopes. We also prove that 2-bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of 2-bridge knots with K_{p'/q'}>K_{p/q} arise from the Ohtsuki-Riley-Sakuma construction. Comment: 24 pages, 4 figures |
| Databáze: | arXiv |
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