Related papers: Id\`elic class field theory for 3-manifolds

We call a knot in the 3-sphere $SU(2)$-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in $SU(2)$ are binary dihedral. This is a generalisation of being a 2-bridge knot.…

We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed…

The first part of this paper is a short review of the construction [dg-ga/9710001] of invariants of rational homology 3-spheres and knots in terms of configuration space integrals. The second part describes the relationship between the…

This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for…

Based on the analogies of arithmetic topology, we show a topological analogue of Hilbert's Satz 90 for idele groups and utilize our previously established Hasse norm principle to present a proof of an Iyanaga--Tamagawa type genus formula…

We prove that for any integer $n$ there exist infinitely many different knots in $S^3$ such that $n$-surgery on those knots yields the same 3-manifold. In particular, when $|n|=1$ homology spheres arise from these surgeries. This answers…

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being…

Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…

In this paper we define and investigate Z/2-homology cobordism invariants of Z/2-homology 3-spheres which turn out to be related to classical invariants of knots. As an application we show that many lens spaces have infinite order in the…

The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic…

The space M_n of all isomorphism classes of n-dimensional Lie algebras over a field k has a natural non-Hausdorff topology, induced from the Segal topology by the action of GL(n). One way of studying this complicated space is by topological…

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…

We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not…

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove…

Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is…

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat…

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form…

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the…

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show…

For a prime knot group, the classifying space for the family of the subgroups generated by the meridians can be seen as an abstract analogue of the ambient manifold in which the knot lives. An explicit model of this ambient classifying…