Classical pretzel knots and left orderability

We consider the classical pretzel knots [Formula: see text], where [Formula: see text] are positive odd integers. By using continuous paths of elliptic [Formula: see text]-representations, we show that (i) the 3-manifold obtained by [Formula: see text]-surgery on [Formula: see text] has left orderable fundamental group if [Formula: see text] and (ii) the [Formula: see text]-cyclic branched cover of [Formula: see text] has left orderable fundamental group if [Formula: see text].

On sofic approximations of

We construct a sofic approximation of ${\mathbb F}_2\times {\mathbb F}_2$ that is not essentially a ‘branched cover’ of a sofic approximation by homomorphisms. This answers a question of L. Bowen.

Complete Classification of Degree 7 for Genus 1

Assume that is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples ( of permutations in the symmetric group , in which and s generate a transitive subgroup G of This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

Signatures of Topological Branched Covers

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma (X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $\Xi (K)$ to $\sigma (X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a topological four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of points on $B$ which are not locally flat, $X$ in this setting is a stratified pseudomanifold, and we use the intersection homology signature of $X$, $\sigma _{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $\Xi (K)$, providing a new technique to potentially detect slice knots that are not ribbon.

Invariants of a General Branched Cover of P1

We investigate the resolution of a general branched cover $\alpha \colon C \to \mathbf{P}^1$ in its relative canonical embedding $C \subset \mathbf{P} E$. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati–Ekedahl bundle, or bundle of quadrics$F$—the 1st bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies $g = 1 \mod d$.

The multi-region index of a knot

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model).

Counting surface branched covers

To a branched cover f between orientable surfaces one can associate a certain branch datum, that encodes the combinatorics of the cover. This satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum there exists a branched cover f such that . One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.

L-spaces, left-orderability and two-bridge knots

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot [Formula: see text] is an L-space and its fundamental group is not left-orderable. Therefore, the family of 3-fold cyclic branched cover of any genus 2 two-bridge knot [Formula: see text] verifies the [Formula: see text]-space conjecture. We also show that if [Formula: see text] is a two-bridge knot with [Formula: see text], [Formula: see text], then the fundamental group of the 5-fold cyclic branched cover of [Formula: see text] is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus 1 two-bridge knot is not left-orderable.

Injectivity of Generalized Wronski Maps

We study linear projections on Plücker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear diòerential operator and the pole placement map for symmetric linear systems are natural examples.