We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Σm(K), and we give computations when m = 2 for over fifty three-bridge knots with up to eleven crossings.

In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on the existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the ...

Consider genus g curves that admit degree d covers of an elliptic curve. Varying a branch point, we get a one-parameter family W of simply branched covers. Varying the target elliptic curve, we get another one-parameter family Y of covers that have a unique branch point. We investigate the geometry of W and Y by using admissible covers to study their slopes, genera and components. The results c...

We study branched coverings in several contexts, proving that under suitable circumstances the cover satisses the same upper curvature bounds as the base space. The rst context is of branched covers of an arbitrary CAT() space over an arbitrary complete convex subset. The second context is of a certain sort of branched cover of a Riemannian manifold over a family of mutually orthogonal submanif...

‎It is shown that the knowledge of a surjective morphism $Xto Y$ of complex‎ ‎curves can be effectively used‎ ‎to make explicit calculations‎. ‎The method is demonstrated‎ ‎by the calculation of $j(ntau)$ (for some small $n$) in terms of $j(tau)$ for the elliptic curve ‎with period lattice $(1,tau)$‎, ‎the period matrix for the Jacobian of a family of genus-$2$ curves‎ ‎complementing the classi...

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers ℙ 2 , with branch locus equal to a curve. This is parametrized by space

We find approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of some other link invariants arising from the homology of branched covers of links.

For each pair of coprime integers n > m ≥ 2 we construct pairs of non equivalent Conway irreducible hyperbolic knots with the same n-fold and m-fold cyclic branched covers. Mathematics Subject Classification (2000). Primary 57M25; Secondary 57M12, 57M50.