Epimorphisms and Boundary Slopes of 2-Bridge Knots

Abstract

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.

24 pages, 4 figures

References in corpus (7)

• Epimorphisms between 2-bridge link groups
• Witten's conjecture and Property P
• On Character varieties of two-bridge knot groups
• Simon's conjecture for 2-bridge knots
• Finiteness of 3-manifolds associated with non-zero degree mappings
• On character varieties, sets of discrete characters, and non-zero degree maps
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Cited by in corpus (6)

• Presentation length and Simon's conjecture
• Tangle sums and factorization of A-polynomials
• Non-meridional epimorphisms of knot groups
• On the Neuwirth conjecture for knots
• Some families of minimal elements for the partial ordering on prime knots
• Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots