Commensurability classes of twist knots

In this paper we prove that if $M_K$ is the complement of a non-fibered twist knot $K$ in $\mathbb S^3$, then $M_K$ is not commensurable to a fibered knot complement in a $\mathbb Z/ 2 \mathbb Z$-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.Comment: 10 pages, 3 figure

Twisted Alexander polynomials of 2-bridge knots

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.Comment: 29 pages, 2 figure

Involutory quandles of (2,2,r)-Montesinos links

In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.Comment: 19 pages, 8 figure

Epimorphisms and Boundary Slopes of 2-Bridge Knots

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 figure

Boundary slopes of 2-bridge links determine the crossing number

A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equal to the crossing number.Comment: 16 pages, 6 figure