Narrow Results

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.

In this work we classify the space of irreducible SU(2) representations of the groups of the twist knots by means of a geometric method which is generalizable to all the 2-bridge knots.

We extend Hoste–Shanahan's calculations for the A-polynomial of twist knots, to give an explicit formula.

We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial ${\mathrm{\Delta }}_0$ (as defined by Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications12 (2003) 987–1000]) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from ${\mathrm{\Delta }}_0$.

Let $M$ be a closed oriented 3-manifold and let $G$ be a discrete group. We consider a representation $\rho :{\pi }_1(M)\to G$. For a 3-cocycle $\alpha$, the Dijkgraaf–Witten invariant is given by $({\rho }^{\ast }\alpha )[M]$, where ${\rho }^{\ast }:H^3(G)\to H^3(M)$ is the map induced by $\rho$, and $[M]$ denotes the fundamental class of $M$. Note that $({\rho }^{\ast }\alpha )[M]=\alpha ({\rho }_{\ast }[M])$, where ${\rho }_{\ast }:H_3(M)\to H_3(G)$ is the map induced by $\rho$, we consider an equivalent invariant ${\rho }_{\ast }[M]\in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\mathrm{\text{SL}}}_2ℂ$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2ℂ)$ to the Bloch group of $ℂ$.

In this paper, by replacing $ℂ$ with a finite field ${𝔽}_p$, we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL${}_2{𝔽}_p$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2{𝔽}_p)$ to the Bloch group of ${𝔽}_p$.