1,628 research outputs found
Algorithmic Invariants for Alexander Modules
Let be a group given by generators and relations. It is
possible to compute a presentation matrix of a module over a ring
through Fox's differential calculus. We show how to use Gröbner
bases as an algorithmic tool to compare the chains of elementary
ideals defined by the matrix. We apply this technique to classical
examples of groups and to compute the elementary ideals of
Alexander matrix of knots up to crossings with the same
Alexander polynomial
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Some analogs of Zariski's Theorem on nodal line arrangements
For line arrangements in P^2 with nice combinatorics (in particular, for
those which are nodal away the line at infinity), we prove that the
combinatorics contains the same information as the fundamental group together
with the meridianal basis of the abelianization. We consider higher dimensional
analogs of the above situation. For these analogs, we give purely combinatorial
complete descriptions of the following topological invariants (over an
arbitrary field): the twisted homology of the complement, with arbitrary rank
one coefficients; the homology of the associated Milnor fiber and Alexander
cover, including monodromy actions; the coinvariants of the first higher
non-trivial homotopy group of the Alexander cover, with the induced monodromy
action.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-28.abs.htm
Lyubeznik numbers of monomial ideals
We study Bass numbers of local cohomology modules supported on squarefree
monomial ideals paying special attention to Lyubeznik numbers. We build a
dictionary between local cohomology modules and minimal free resolutions that
allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of
the linear strands of the Alexander dual ideals. The methods we develop also
help us to give a bound for the injective dimension of the local cohomology
modules in terms of the dimension of the small support.Comment: 28 page
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
Faithful Lie algebra modules and quotients of the universal enveloping algebra
We describe a new method to determine faithful representations of small
dimension for a finite dimensional nilpotent Lie algebra. We give various
applications of this method. In particular we find a new upper bound on the
minimal dimension of a faithful module for the Lie algebras being counter
examples to a well known conjecture of J. Milnor
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