68 research outputs found
Enumeration of the BranchedmZp-Coverings of Closed Surfaces
AbstractIn this paper, we enumerate the equivalence classes of regular branched coverings of surfaces whose covering transformation groups are the direct sum of m copies of Zp, p prime
The concordance genus of knots
In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K):
these are the classical genus, the 4-ball genus, and the concordance genus,
defined to be the minimum genus among all knots concordant to K. Clearly 0 <=
g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that
g_4(K) need not equal g_c(K). We begin by reviewing and extending their
results.
For knots representing elements in A, the concordance group of algebraically
slice knots, the relationships between these genera are less clear. Casson and
Gordon's result that A is nontrivial implies that g_4(K) can be nonzero for
knots in A. Gilmer proved that g_4(K) can be arbitrarily large for knots in A.
We will prove that there are knots K in A with g_4(K) = 1 and g_c(K)
arbitrarily large.
Finally, we tabulate g_c for all prime knots with 10 crossings and, with two
exceptions, all prime knots with fewer than 10 crossings. This requires the
description of previously unnoticed concordances.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-1.abs.htm
Spin Hurwitz numbers and topological quantum field theory
Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the
size of their automorphism group (like ordinary Hurwitz numbers), but signed
according to the parity of the covering surface. These numbers were
first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of
holomorphic differentials on a Riemann surface. They have also been related to
Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and
Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes
these numbers, and deduce a formula for any genus in terms of the combinatorics
of the Sergeev algebra, generalizing the formula of
Eskin-Okounkov-Pandharipande. During the construction, we describe a procedure
for averaging any TQFT over finite covering spaces based on the finite path
integrals of Freed-Hopkins-Lurie-Teleman.Comment: 38 pages, substantially rewritten and reorganized following referees'
advice. Diagrams added. The key results are unchange
Algorithmic aspects of branched coverings
This is the announcement, and the long summary, of a series of articles on
the algorithmic study of Thurston maps. We describe branched coverings of the
sphere in terms of group-theoretical objects called bisets, and develop a
theory of decompositions of bisets.
We introduce a canonical "Levy" decomposition of an arbitrary Thurston map
into homeomorphisms, metrically-expanding maps and maps doubly covered by torus
endomorphisms. The homeomorphisms decompose themselves into finite-order and
pseudo-Anosov maps, and the expanding maps decompose themselves into rational
maps.
As an outcome, we prove that it is decidable when two Thurston maps are
equivalent. We also show that the decompositions above are computable, both in
theory and in practice.Comment: 60-page announcement of 5-part text, to apper in Ann. Fac. Sci.
Toulouse. Minor typos corrected, and major rewrite of section 7.8, which was
studying a different map than claime
Cut-and-join structure and integrability for spin Hurwitz numbers
Spin Hurwitz numbers are related to characters of the Sergeev group, which
are the expansion coefficients of the Q Schur functions, depending on odd times
and on a subset of all Young diagrams. These characters involve two dual
subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur
functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators
W_\Delta with \Delta\in OP. The eigenvalues of these operators are the
generalized Sergeev characters, their algebra is isomorphic to the algebra of Q
Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the
generating function of spin Hurwitz numbers is a \tau-function of an integrable
hierarchy, that is, of the BKP type. At last, we discuss relations of the
Sergeev characters with matrix models.Comment: 22 page
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
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