In this thesis we construct an additive category whose objects are embedded graphs (or in particular knots) in the 3-sphere and where morphisms are formal linear combinations of 3-manifolds. Our definition of correspondences relies on the Alexander branched covering theorem [1]， which shows that all compact oriented 3-manifolds can be realized as branched coverings of the 3-sphere， with branched locus an embedded (not necessarily connected) graph. The way in which a given 3-manifold is realized as a branched cover is highly not unique. It is precisely this lack of uniqueness that makes it possible to regard 3-manifolds as correspondences. In fact， we show that， by considering a 3-manifold M realized in two different ways as a covering of the 3-sphere as defining a correspondence between the branch loci of the two covering maps， we obtain a well defined associative composition of correspondences given by the fibered product.
目錄：
1.Introduction
Chapter 1.Graphs Category and Three-manifolds as correspondences
1.Three-manifolds as correspondences
2.Composition ofcorrespondencas
3.Representations and compositions of correspondences
4.Semigroupoids and additive categories
5.Categories of graphs and correspondences
6.Convolution algebra and time evolution
7.Equivalence of correspondences
8.Convolution algebras and 2-semigroupoids
9.Vertical and horizontal time evolutions
10.Vertical time evolution: Hattie-Hawking gravity
11.Vertical time evolution: gauge moduli and index theory
12.Horizontal time evolution: bivariant Chern character
13.Noncommutative spaces and spectral correspondences
Chapter 2.Knots， Khovanov Homology
1.Introduction
2.From graphs to knots
3.Khovanov Homology
4.Knots and Links Cobordism Groups
5.Graphs and cobordisms
6.Homology theories for embedded graphs
7.Questions and Future Work
Appendix A.
1.Branched Covering
2.Filtration
3.Knot and link
4.Topological Quantum Field Theory
5.2-Category
6.Group Rings
7.Creation and annihilation operators
8.A quick introduction to Dirac operators
9.Concepts of Cyclic Cohomology
Appendix.Bibliography
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