78 research outputs found
Applying TQFT to count regular coverings of Seifert 3-manifolds
I give a formula for computing the number of regular -coverings of
closed orientable Seifert 3-manifolds, for a given finite group . The
number is computed using a 3d TQFT with finite gauge group, through a
cut-and-glue process.Comment: 16 pages; Journal of Geometry and Physics, 201
Explicit computation of some families of Hurwitz numbers
We compute the number of (weak) equivalence classes of branched covers from a
surface of genus g to the sphere, with 3 branching points, degree 2k, and local
degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2),
(d_1,...,d_m), for several values of g and h. We obtain explicit formulae of
arithmetic nature in terms of the d_i's. Our proofs employ a combinatorial
method based on Grothendieck's dessins d'enfant.Comment: To appear in European Journal of Combinatorics (2018); 23 pages, 12
figure
Counting surface branched covers
To a branched cover f between orientable surfaces one can associate a certain
branch datum D(f), that encodes the combinatorics of the cover. This D(f)
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 D there exists a branched cover f such that D(f)=D. 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. And it turns out that quite
a few natural choices 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 we employ Grothendieck's
dessins d'enfant.Comment: 15 pages, 1 figur
Explicit computation of some families of Hurwitz numbers, II
We continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface,
3 branching points, degree 2k, and local degrees over the branching points of the form [2,...,2], [2h+1,3,2,...,2], [d_1,...,d_L]. We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the d_i's
- …