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Stavros Garoufalidis

Publications and source records attributed to Stavros Garoufalidis.

157 records · Page 9Linked to original sources
arXiv1999

The Aarhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S^3

Path integrals don't really exist, but it is very useful to dream that they do exist, and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S^3 don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology spheres. We show that this invariant is equal to the LMO (Le-Murakami-Ohtsuki) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4-part series, containing the introductions and answers to some frequently asked questions. Theorems are stated but not proved in this part, and it can be viewed as a "research announcement". Part II of this series is titled "Invariance and Universality" (see math/9801049), part III "The Relation with the Le-Murakami-Ohtsuki Invariant" (see math/9808013), and part IV "The Relation with the Rozansky and Ohtsuki Invariants".

q-alg↗
arXiv1998

Signatures of links and finite type invariants of cyclic branched covers

Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational homology 3-sphere. Using elementary principles, we provide a similar calculation for the general case. In addition, we calculate the LMO invariant of the p-fold branched cover of twisted knots in S^3 in terms of the Kontsevich integral of the knot.

math.GT↗
arXiv1998

Values of zeta functions at negative integers, Dedekind sums and toric geometry

This is an expanded version. We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of the above mentioned relations and explicit computations of the various zeta values and Dedekind sums involved.

alg-geom↗
arXiv1998

Wheels, Wheeling, and the Kontsevich Integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of Chinese characters. The two formulas use the related notions of "Wheels" and "Wheeling". We prove these formulas "on the level of Lie algebras" using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras.

q-alg↗
arXiv1997

Finite type invariants, the mapping class group and blinks

The goal of the present paper is to find higher genus surgery formulae for the set of finite-type invariants of homology spheres, and to develop a companion theory of finite-type invariants to be applied, in a subsequent publication, to the study of subgroups of the mapping class group. The main result is to show that six filtrations on the vector space generated by oriented homology spheres (three coming from surgery on special classes of links and three coming from subgroups of the mapping class group) are equal. En route we introduce the notion of blink (a special case of a link) and of a new subgroup of the mapping class group.

q-alg↗
arXiv1997

Applications of the lantern identity

The purpose of this note is to unify the role of the lantern identity in the proof of several finiteness theorems. In particular, we show that for every nonnegative integer m, the vector space (over the rationals) of type m (resp. T-type m) invariants of integral homology 3-spheres are finite dimensional. These results have already been obtained by [Oh] and [GL2] respectively; our derivation however is simpler, conceptual and relates to several other applications of the lantern identity.

q-alg↗
arXiv1997

Finite type 3-manifold invariants and the structure of the Torelli group I

Using the recently developed theory of finite type invariants of integral homology 3-spheres we study the structure of the Torelli group of a closed surface. Explicitly, we construct (a) natural cocycles of the Torelli group (with coefficients in a space of trivalent graphs) and cohomology classes of the abelianized Torelli group; (b) group homomorphisms that detect (rationally) the nontriviality of the lower central series of the Torelli group. Our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the Casson invariant in the theory of finite type invariants of integral homology 3-spheres and in Morita's study of the structure of the Torelli group Our results generalize those of S. Morita {Mo1,Mo2} and complement the recent calculation, due to R. Hain, of the I-adic completion of the rational group ring of the Torelli group. We also give analogous results for two other subgroups of the mapping class group.

q-alg↗
arXiv1997

Finite type invariants III: manifold weight systems

We introduce the notion of weight system for finite type invariants of integral homology 3-spheres, and we show that invariants of type m are determined, modulo invariants of type m-1, by their associated weight system.

q-alg↗
arXiv1996

Finite type 3-manifold invariants and the structure of the Torelli

We apply the theory of finite-type invariants of homology 3-spheres to investigate the structure of the Torelli group. We construct natural cocycles in the Torelli group and show that the lower central series quotients of the Torelli group map onto a vector space of trivalent graphs. We also have analogous results for two other natural subgroups of the mapping class group.

q-alg↗
arXiv1995

On Finite Type 3-manifold invariants IV: Comparison of Definitions

This paper compares the definitions of finite-type invariants due to Ohtsuki and to Garoufalidis, showing that, residually, type 3m of the former equals type m of the latter. It also shows that type 2m Ohtsuki invariants define knot invariants of type 3m (first proved by Habegger).

q-alg↗
arXiv1995

On Finite Type 3-Manifold Invariants II

This paper continues the study of finite-type invariants of homology spheres studied by Ohtsuki and Garoufalidis. We apply the surgery classification of links to give a diagrammatic description, using ideas of Ohtsuki. This uses a computation of the surgery equivalence classes of pure braids. We show that the order of any invariant, in Ohtsukis sense, is a multiple of 3. We also study the relation between the order of an invariant and that of the knot invariant it defines.

q-alg↗