A locally minimal, but not globally minimal bridge position of a knot

We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.Comment: 27 pages, 12 figures, v3: minor corrections throughout the pape

Cannon-Thurston Maps for Kleinian Groups

We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.Comment: 39 pgs 1 fig. Final version incorporating referee comments. To appear in Forum of Mathematics, P

Non-minimal bridge positions of torus knots are stabilized

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.Comment: 11 pages, 4 figure

Mutation and SL(2,C)-Reidemeister torsion for hyperbolic knots

Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift of the holonomy to SL(2,C) is invariant under mutation along a Conway sphere.Comment: 19 pages, 1 figur

Invariant solutions to the Strominger system and the heterotic equations of motion

We construct many new invariant solutions to the Strominger system with respect to a 2-parameter family of metric connections $\nabla^{\varepsilon,\rho}$ in the anomaly cancellation equation. The ansatz $\nabla^{\varepsilon,\rho}$ is a natural extension of the canonical 1-parameter family of Hermitian connections found by Gauduchon, as one recovers the Chern connection $\nabla^{c}$ for $({\varepsilon,\rho})=(0,\frac12)$, and the Bismut connection $\nabla^{+}$ for $({\varepsilon,\rho})=(\frac12,0)$. In particular, explicit invariant solutions to the Strominger system with respect to the Chern connection, with non-flat instanton and positive $\alpha'$ are obtained. Furthermore, we give invariant solutions to the heterotic equations of motion with respect to the Bismut connection. Our solutions live on three different compact non-K\"ahler homogeneous spaces, obtained as the quotient by a lattice of maximal rank of a nilpotent Lie group, the semisimple group SL(2,$\mathbb{C}$) and a solvable Lie group. To our knowledge, these are the only known invariant solutions to the heterotic equations of motion, and we conjecture that there is no other such homogeneous space admitting an invariant solution to the heterotic equations of motion with respect to a connection in the ansatz $\nabla^{\varepsilon,\rho}$.Comment: 27 pages, 3 figure

Chimneys, leopard spots, and the identities of Basmajian and Bridgeman

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function q_n of degree 2(n-2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity \chi(S) = \sum_i q_n(e^{l_i}) where the sum is taken over the orthospectrum of M. When n=3, this has the explicit form \sum_i 1/(e^{2l_i}-1) = -\chi(S)/4.Comment: 6 pages; version 2 incorporates referee's comment