13,858 research outputs found
Three-dimensional simplicial gravity and combinatorics of group presentations
We demonstrate how some problems arising in simplicial quantum gravity can be
successfully addressed within the framework of combinatorial group theory. In
particular, we argue that the number of simplicial 3-manifolds having a fixed
homology type grows exponentially with the number of tetrahedra they are made
of. We propose a model of 3D gravity interacting with scalar fermions, some
restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix
model. We propose a qualitative picture of the phase structure of 3D simplicial
gravity compatible with the numerical experiments and available analytical
results.Comment: 24 page
Milnor invariants and the HOMFLYPT polynomial
We give formulas expressing Milnor invariants of an n-component link L in the
3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor
invariant \bar{\mu}_J(L) vanishes for any sequence J with length at most k,
then any Milnor \bar{\mu}-invariant \bar{\mu}_I(L) with length between 3 and
2k+1 can be represented as a combination of HOMFLYPT polynomial of knots
obtained from the link by certain band sum operations. In particular, the
`first non vanishing' Milnor invariants can be always represented as such a
linear combination.Comment: Entirely revised version (20 pages). The main result was generalized
and extended to Milnor invariants of links, using new arguments. Several
corollaries are given, in particular one containing the main result of the
previous version. Example and References adde
Using groups for investigating rewrite systems
We describe several technical tools that prove to be efficient for
investigating the rewrite systems associated with a family of algebraic laws,
and might be useful for more general rewrite systems. These tools consist in
introducing a monoid of partial operators, listing the monoid relations
expressing the possible local confluence of the rewrite system, then
introducing the group presented by these relations, and finally replacing the
initial rewrite system with a internal process entirely sitting in the latter
group. When the approach can be completed, one typically obtains a practical
method for constructing algebras satisfying prescribed laws and for solving the
associated word problem
A finer reduction of constraint problems to digraphs
It is well known that the constraint satisfaction problem over a general
relational structure A is polynomial time equivalent to the constraint problem
over some associated digraph. We present a variant of this construction and
show that the corresponding constraint satisfaction problem is logspace
equivalent to that over A. Moreover, we show that almost all of the commonly
encountered polymorphism properties are held equivalently on the A and the
constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture
as well as the conjectures characterizing CSPs solvable in logspace and in
nondeterministic logspace are equivalent to their restriction to digraphs.Comment: arXiv admin note: substantial text overlap with arXiv:1305.203
COSMICAH 2005: workshop on verification of COncurrent Systems with dynaMIC Allocated Heaps (a Satellite event of ICALP 2005) - Informal Proceedings
Lisboa Portugal, 10 July 200
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