17 research outputs found
Equations at infinity for critical-orbit-relation families of rational maps
We develop techniques for using compactifications of Hurwitz spaces to study
families of rational maps defined by critical
orbit relations. We apply these techniques in two settings: We show that the
parameter space of degree- bicritical maps with a
marked 4-periodic critical point is a -punctured Riemann surface of genus
. We also show that the parameter space
of degree-2 rational maps with a marked 5-periodic
critical point is a 10-punctured elliptic curve, and we identify its
isomorphism class over . We carry out an experimental study of the
interaction between dynamically defined points of (such as
PCF points or punctures) and the group structure of the underlying elliptic
curve.Comment: Significant revisions and generalizations, added new application
(Section 5), 22 page
Genus-zero -spin theory
We provide an explicit formula for all primary genus-zero -spin
invariants. Our formula is piecewise polynomial in the monodromies at each
marked point and in . To deduce the structure of these invariants, we use a
tropical realization of the corresponding cohomological field theories.Comment: 28 page
On product identities and the Chow rings of holomorphic symplectic varieties
For a moduli space of stable sheaves over a surface , we propose
a series of conjectural identities in the Chow rings generalizing the classic Beauville-Voisin identity for
a surface. We emphasize consequences of the conjecture for the structure
of the tautological subring The conjecture
places all tautological classes in the lowest piece of a natural filtration
emerging on , which we also discuss. We prove the proposed
identities when is the Hilbert scheme of points on a surface.Comment: A discussion of a natural filtration on the Chow groups of M,
emerging from the paper's point of view, was adde
Planar and spherical stick indices of knots
The stick index of a knot is the least number of line segments required to
build the knot in space. We define two analogous 2-dimensional invariants, the
planar stick index, which is the least number of line segments in the plane to
build a projection, and the spherical stick index, which is the least number of
great circle arcs to build a projection on the sphere. We find bounds on these
quantities in terms of other knot invariants, and give planar stick and
spherical stick constructions for torus knots and for compositions of trefoils.
In particular, unlike most knot invariants,we show that the spherical stick
index distinguishes between the granny and square knots, and that composing a
nontrivial knot with a second nontrivial knot need not increase its spherical
stick index
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure