Patrick D. Shanahan

A FORMULA FOR THE A-POLYNOMIAL OF TWIST KNOTS

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.

TRACE FIELDS OF TWIST KNOTS

In this paper we compute the trace field for the family of hyperbolic twist knots. We describe this field as a simple extension ℚ(z0) where z0 is a specified root of a particular irreducible polynemial Φn(z)∈ℤ[z]. As a consequence, we find that the degree of the trace field is precisely two less than the-minimal crossing number of a twist knot.

Epimorphisms and boundary slopes of 2–bridge knots

In this article we study a partial ordering on knots in S 3 where K1 K2 if there is an epimorphism from the knot group of K1 onto the knot group of K2 which preserves peripheral structure. If K1 is a 2‐bridge knot and K1 K2 , then it is known that K2 must also be 2‐bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2‐bridge knot Kp=q , produces infinitely many 2‐bridge knots Kp0=q0 with Kp0=q0 Kp=q . After characterizing all 2‐bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, Kp0=q0 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 Kp0=q0 Kp=q arise from the Ohtsuki‐Riley‐Sakuma construction. 57M25

COMMENSURABILITY CLASSES OF TWIST KNOTS

In this paper we prove that if MK is the complement of a non-fibered twist knot K in

Computing Boundary Slopes of 2-Bridge Links

\mathbb S^3

TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS

, then MK is not commensurable to a fibered knot complement in a ℤ/2ℤ-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 Calegari and Dunfield is satisfied for these varieties. In addition, we partially extend our results to a second infinite family of 2-bridge knots.

Cyclic Dehn surgery and the A-polynomial

We describe an algorithm for computing boundary slopes of 2bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.

Three-Dimensional Representations of Punctured Torus Bundles

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.

Links with finite n–quandles

Abstract We present a necessary condition for Dehn surgery on a knot in S 3 to be cyclic which is based on the A-polynomial of the knot. The condition involves a width of the Newton polygon of the A-polynomial, and provides a simple method of computing a list of possible cyclic surgery slopes. The width produces a list of at most three slopes for a hyperbolic knot which contains no closed essential surface in its complement (in agreement with the Cyclic Surgery Theorem). We conclude with an application to cyclic surgeries along non-boundary slopes of hyperbolic mutant knots.

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

In this paper, we construct a complex curve of irreducible representations of the fundamental group of a once punctured torus bundle over the circle. These representations are different from those obtained by composing representations in with the unique irreducible representation of in . Moreover, infinitely many of these representations are conjugate to SU(3) representations. We conclude the paper with a computation of the curve in the case that the bundle is the figure-eight knot complement, and we show that for infinitely many Dehn surgeries on the figure-eight knot, there is a representation from this curve that descends to a representation of the fundamental group of the surgered manifold.