Author(s): Kuperberg, Greg; Thurston, Dylan P. | Abstract: We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.
Two other generalizations that can be considered are invariants of graphs in 3-manifolds, and invariants associated to other flat connections @cite_16 . We will analyze these in future work. Among other things, there should be a general relation between flat bundles and links in 3-manifolds on the one hand and finite covers and branched covers on the other hand @cite_26 .
@cite_16 @cite_26
2080743555 1641082372 1990891135 2023137590
The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory 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. We describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature. Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs (Z H ) in a Calabi–Yau threefold (X ). Here (H ) is a member of a sufficiently positive linear system and (Z ) is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of (Z H ), pushed forward to (X ) and considered as a certain Joyce–Song pair in the derived category of (X ). We express these invariants in terms of the MNOP invariants of (X ).