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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 49021 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Moghaddasi, Reza; Eftekhari, Eaman; Daemi, Ali Akbar
- Abstract:
- Knot theory is the study of ambient isotopy classes of compact 1–manifolds in a 3-manifold. In classical knot theory this 3-manifold is R3 or S3. This field has experienced a great transformative advances in recent years because of its strong connections with and a number of other mathematical disciplines including topology of 3-manifolds and 4-manifolds, gauge theory, representation theory, categorification, morse theory, symplectic geometry and the theory of pseudo-holomorphic curves. In this thesis we start with classical knot theory, introducing some of its (classical) invariants like unknotting number, Seifert genus and slice genus of a knot, knot group and finally Alexander Polynomial of knots. In the following we will study a homological invariant in knot thery known as Knot Floer Homology (Grid Homology) from a combinatorial viewpoint. This invariant which can be understood as a ctagorification of Alexander polynomial contains very useful information about knots. At the end we will see some of its topological applications including a proof for Milnor’s conjecture about Torus knots and the existence of exotic structures on R4
- Keywords:
- Knot Theory ; Grid Homology ; Knot Floar Homology ; Alexander Polynomial ; T-invariant ; Exotic Structures on R4
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