mm-4139
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Subject: Paper published by Geometry and Topology
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
The following paper has been published:
Geometry and Topology, Volume 9 (2005) Paper no. 36, pages 1603--1637
URL:
http://www.maths.warwick.ac.uk/gt/GTVol9/paper36.abs.html
DOI: 10.2140/gt.2005.9.1603
Title:
Knot and braid invariants from contact homology II,
Author(s):
Lenhard Ng
with an appendix written jointly with Siddhartha Gadgil
Abstract:
We present a topological interpretation of knot and braid contact
homology in degree zero, in terms of cords and skein relations. This
interpretation allows us to extend the knot invariant to embedded
graphs and higher-dimensional knots. We calculate the knot invariant
for two-bridge knots and relate it to double branched covers for
general knots.
In the appendix we show that the cord ring is determined by the
fundamental group and peripheral structure of a knot and give
applications.
Secondary: 53D35, 20F36
Keywords:
Contact homology, knot invariant, differential graded algebra, skein
relation, character variety
Received: 24 February 2005
Accepted: 16 August 2005
Published: 26 August 2005
Proposed: Yasha Eliashberg
Seconded: Robion Kirby, Ronald Fintushel
Author(s) address(es):
LN: Department of Mathematics, Stanford University
Stanford, CA 94305, USA
SG: Stat-Math Unit, Indian Statistical Institute
Bangalore, India
Email: lng@math.stanford.edu, gadgil@isibang.ac.in
URL: http://math.stanford.edu/~lng/
Subject: A Category of Godel Codings
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
I have been studying basic recursion/computability theory lately, and
one thing in particular has been troubling me. It is commonly pointed
out, in textbooks and elsewhere, that we don't lose much by focusing
our study of computability to only the set of natural numbers. After
all, following the example of Godel, we just equate the discrete set of
our study with a certain subset of the natural numbers via an
intuitively effective map, and then proceed from there. For most
purposes, this is well enough, especially as a theoretical
simplification.
Let me make an analogy. Assuming the axiom of choice, every vector
space V (over R, say) has a basis. In this way we may equate V with a
vector space with is the direct sum of R dim V times. However,
spaces over R as the study of such direct sums. Likewise, it seems
somewhat disingenuous to equate the study of computability on discrete
structures with computability on N, because in like manner, we must
make an arbitary choice to associate the discrete set in question with
N.
For example, suppose I want to choose a coding method for, say, graphs
with rationally weighted edges. In computer science parlance, I need a
'file format.' [Note a file is little more than a element of N padded
by zeros.] Once I know the existence of such a coding, constructing a
coding is as much an exercise in engineering and design as it is
mathematics. To be brief, the process of coding discrete objects
requires subjective choices and is highly non-canonical. This lack of
universality is somewhat disturbing.
I have spent many hours thinking about this and related issues, finally
I came up with the following construction. On further reflection, I am
extremely surprised I have not seen it yet. The construction itself is
very simple. My experience in foundations is limited, so I would
appreciate any comments.
The construction goes as follows:
[Note that using 'recursive' instead of 'primitive recursive' below
results in a rather boring theory]
Definition: A primitive recursive type, or a primitive, is a pair
(A,~) where Aand ~ are unary and binary function symbols in Primitive
Recursive Arithmetic respectively, such that PRA proves A and ~ are
characteristic functions and ~ induces a equivalence relation.
Notationally we use A and ~ as if they were sets/relations. Given
primitives (A,~) and (B,~) and a unary function f in PRA, we saythat f
induces a map from A to B if PRA proves
[Which equivalence ~ denotes will be clear from the context]
If f and g induce a map from A to B, and PRA proves
then we write f /simeq g.
For each primitive A there is an obvious identity map 1_A.
This gives us a category PRT whose objects are primitives (A,~) and
functions which induce maps from A to B, modulo /simeq.
If we wish, we can now consider the purely categorical properties of
PRT, and thus avoid the issue of canonicalism completely. Note each
object in PRT has exactly countably many isomorphic object (Replace A
with 2^A,and modify ~ appropiately, etc...).
The isomorphism problem for this category is unsolvable, as follows:
Consider any Pi_1 statement B, which we can represent as ForAll x A(x)
where A(x)is primitive recursive. Let (A',=) be the corresponding
primitive, where A' is the unary function symbol representing A and =,
to abuse notation, is the characteristic function of equality. Define
identity function on N is an isomorphism between A' and A'' in the
category PRT if and only if ForAll x A(x), i.e. if and only if B holds.
Since B was arbitrary, this shows the isomorphism problem in PRT is
undecidable.
After a few definitions, there is wealth of primitives we can construct
quickly
First off, by considering regular equality, there is a primitive for
everything primitive recursive subset of N.
Given primitives A and B we can construct, using standard codings:
1. A x B: The Cartesian and categorical product of A and B,
2. A + B: The disjoint union/ categorical sum of A and B,
3. A^*: The primitive of finite A-sequences
4. P_f(A): The primitive of finite subsets of A
6. A x ~~: Given (A,~), and a primitive recursive equivalence ~~,
consider the equivalence x % y iff x ~ y and x ~~ y. This comes
Construction 6 is especially important, as it is a sort of separation
principle.
Using these constructions we can describe primitives for just about
every conceivable combinatorial structure, labeled or unlabeled.
Immediately a whole slew of questions come to mind.
1. Can we axiomatize PRT via its categorical properties?
2. In general, in what other ways can we specify PRT?
2. Can we find a reasonable formal system for PRT, the 'universe' of
primitives?
2. What if we replace the use of 'primitive recursive' above with
Delta_0 in PA or with various complexity classes like NP and P? The
Isomorphism problem in these categories corresponds to what?
Rex Butler
RexButler@hotmail.com
Subject: Re: thousand of books to download
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Surely this is massive copyright violation? Many of these books are
undoubtedly out of copyright, but many of them are very obviously still
in copyright. The site is Russian; but I thought that Russia now
abided by the Berne copyright convention?
--Ron Bruck
Subject: Two papers published by Geometry and Topology
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
The following two papers have been published.
(1)
K- and L-theory of the semi-direct product of the discrete
3-dimensional Heisenberg group by Z/4
by
Wolfgang Lueck
DOI: 10.2140/gt.2005.9.1639
URL: http://www.maths.warwick.ac.uk/gt/GTVol9/paper37.abs.html
(2)
Strongly fillable contact 3-manifolds without Stein fillings
by
Paolo Ghiggini
DOI: 10.2140/gt.2005.9.1677
URL: http://www.maths.warwick.ac.uk/gt/GTVol9/paper38.abs.html
Details follow:
(1)
Geometry and Topology, Volume 9 (2005) Paper no. 37, pages 1639--1676
URL:
http://www.maths.warwick.ac.uk/gt/GTVol9/paper37.abs.html
DOI: 10.2140/gt.2005.9.1639
Title:
K- and L-theory of the semi-direct product of the discrete 3-dimensional
Heisenberg group by Z/4
Author(s):
Wolfgang Lueck
Abstract:
We compute the group homology, the topological K-theory of the reduced
C^*-algebra, the algebraic K-theory and the algebraic L-theory of the
group ring of the semi-direct product of the three-dimensional
discrete Heisenberg group by Z/4. These computations will follow from
the more general treatment of a certain class of groups G which occur
which satisfies certain assumptions. The key ingredients are the
Baum-Connes and Farrell-Jones Conjectures and methods from equivariant
algebraic topology.
Secondary: 19A31, 19B28, 19D50, 19G24, 55N99
Keywords:
K- and L-groups of group rings and group C^*-algebras, three-dimensional
Heisenberg group.
Accepted: 19 August 2005
Published: 28 August 2005
Proposed: Gunnar Carlsson
Seconded: Ralph Cohen, Bill Dwyer
Author(s) address(es):
Fachbereich Mathematik, Universitaet Muenster
Einsteinstr. 62, 48149 Muenster, Germany
Email: lueck@math.uni-muenster.de
URL: www.math.uni-muenster.de/u/lueck/
(2)
Geometry and Topology, Volume 9 (2005) Paper no. 38, pages 1677--1687
URL:
http://www.maths.warwick.ac.uk/gt/GTVol9/paper38.abs.html
DOI: 10.2140/gt.2005.9.1677
Title:
Strongly fillable contact 3-manifolds without Stein fillings
Author(s):
Paolo Ghiggini
Abstract:
We use the Ozsvath-Szabo contact invariant to produce examples of
strongly symplectically fillable contact 3-manifolds which are not
Stein fillable.
Secondary: 57R57
Keywords:
Contact structure, symplectically fillable, Stein fillable, Ozsvath-Szabo
invariant
Received: 23 June 2005
Accepted: 4 August 2005
Published: 28 August 2005
Proposed: Peter Ozsvath
Seconded: Robion Kirby, Yasha Eliashberg
Author(s) address(es):
CIRGET, Universite du Quebec a Montreal
Case Postale 8888, succursale Centre-Ville
Montreal (Quebec) H3C 3P8, Canada
Email: ghiggini@math.uqam.ca
Subject: Volterra integral equation of the first kind
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Consider the integral equation:
/ x
| K(x,t)*theta(t)dt = f(x)
/ a
where the following are true:
a = -4
-4 < x < 0
K(x,t) = x + t
and
f(x) = 1
Two questions:
1) does this equation fit the definition of a
Volterra integral equation of the first kind?
2) what is theta(t)? (or can it be proven that given the above
conditions
theta(t) does not exist?)
Subject: A5 cases for the quintic
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
I am debugging a code to calculate Galois groups for quintic equations.
The A5 case x^5 + 20x - 16, is that the case with the smallest integer
coefficients? (I can allow more terms than in this equation.)
Kent Holing