681 research outputs found
Matrix factorizations and link homology
For each positive integer n the HOMFLY polynomial of links specializes to a
one-variable polynomial that can be recovered from the representation theory of
quantum sl(n). For each such n we build a doubly-graded homology theory of
links with this polynomial as the Euler characteristic. The core of our
construction utilizes the theory of matrix factorizations, which provide a
linear algebra description of maximal Cohen-Macaulay modules on isolated
hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep
Normal Domains Arising from Graph Theory
Determining whether an arbitrary subring R of k[x1±1,...,xn±1] is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. First, we determine normality in the case where R is a monomial generated domain where the generators have the form (xixj)±1. Using results for this special case we generalize to the case when R is a monomial generated domain where the generators have the form xi±1xj±1. In both cases, for the ring R, we consider the combinatorial structure that assigns an edge in a mixed directed signed graph to each monomial of the ring. We then use this relationship to provide a combinatorial characterization of the normality of R, and, when R is not normal, we use the combinatorial characterization to compute the normalization of R. Using this construction, we also determine when the ring R satisfies Serre\u27s R1 condition. We also discuss generalizations of this to directed graphs with a homogenizing variable and a special class of hypergraphs
International Journal of Mathematical Combinatorics, Vol.7A
The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences
Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
We provide three 3-dimensional characterizations of the Z-slice genus of a
knot, the minimum genus of a locally-flat surface in 4-space cobounding the
knot whose complement has cyclic fundamental group: in terms of balanced
algebraic unknotting, in terms of Seifert surfaces, and in terms of
presentation matrices of the Blanchfield pairing. This result generalizes to a
knot in an integer homology 3-sphere and surfaces in certain simply connected
signature zero 4-manifolds cobounding this homology sphere. Using the
Blanchfield characterization, we obtain effective lower bounds for the Z-slice
genus from the linking pairing of the double branched cover of the knot. In
contrast, we show that for odd primes p, the linking pairing on the first
homology of the p-fold branched cover is determined up to isometry by the
action of the deck transformation group on said first homology.Comment: 39 pages, 5 figures, comments are welcome! v2: Added generalization
of the main theorem to knots and surfaces in more general 3- and 4-manifolds;
added new corollary showing equality of the Z-slice genus and the superslice
genus; expanded introduction, and added example in last sectio
Twisty itsy bitsy topological field theory
We extend the topological field theory (``itsy bitsy topological field
theory"') of our previous work from mod-2 to twisted coefficients. This
topological field theory is derived from sutured Floer homology but described
purely in terms of surfaces with signed points on their boundary (occupied
surfaces) and curves on those surfaces respecting signs (sutures). It has
information-theoretic (``itsy'') and quantum-field-theoretic (``bitsy'')
aspects. In the process we extend some results of sutured Floer homology,
consider associated ribbon graph structures, and construct explicit admissible
Heegaard decompositions.Comment: 52 pages, 26 figure
Root and weight semigroup rings for signed posets
We consider a pair of semigroups associated to a signed poset, called the
root semigroup and the weight semigroup, and their semigroup rings,
and , respectively.
Theorem 4.1.5 gives generators for the toric ideal of affine semigroup rings
associated to signed posets and, more generally, oriented signed graphs. These
are the subrings of Laurent polynomials generated by monomials of the form
. This result appears to be new
and generalizes work of Boussicault, F\'eray, Lascoux and Reiner, of Gitler,
Reyes, and Villarreal, and of Villarreal. Theorem 4.2.12 shows that strongly
planar signed posets have rings ,
which are complete intersections, with
Corollary 4.2.20 showing how to compute in this case. Theorem 5.2.3
gives a Gr\"obner basis for the toric ideal of in type B,
generalizing Proposition 6.4 of F\'eray and Reiner. Theorems 5.3.10 and 5.3.1
give two characterizations (via forbidden subposets versus via inductive
constructions) of the situation where this Gr\"obner basis gives a complete
intersection presentation for its initial ideal, generalizing Theorems 10.5 and
10.6 of F\'eray and Reiner.Comment: 170 pages; 63 figures; PhD Dissertation, University of Minnesota,
August 201
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
- …