Combinatorial Knot Floer Homology
Author : Divya Sukumaran Nair
Publisher :
Page : 100 pages
File Size : 13,90 MB
Release : 2012
Category : Electronic books
ISBN :
Grid Homology for Knots and Links
Author : Peter S. Ozsváth
Publisher : American Mathematical Soc.
Page : 423 pages
File Size : 33,36 MB
Release : 2015-12-04
Category : Education
ISBN : 1470417375
Book Description
Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.
Studying Surfaces in 4-dimensional Space Using Combinatorial Knot Floer Homology
Author : Matthew Graham
Publisher :
Page : 136 pages
File Size : 43,80 MB
Release : 2012
Category : Homology theory
ISBN :
Book Description
Bordered Heegaard Floer Homology
Author : Robert Lipshitz
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 11,11 MB
Release : 2018-08-09
Category : Mathematics
ISBN : 1470428881
Book Description
The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ^HF of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ^HF. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.
Grid Homology for Knots and Links
Author : Peter Steven Ozsvâath
Publisher :
Page : 410 pages
File Size : 35,51 MB
Release : 2015
Category : MATHEMATICS
ISBN : 9781470427399
Book Description
Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology.
Grid Homology for Knots and Links
Author : Peter S. Ozsvath
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 46,63 MB
Release : 2017-01-19
Category : Education
ISBN : 1470434423
Book Description
Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.
Combinatorial Floer Homology
Author : Vin de Silva
Publisher : American Mathematical Soc.
Page : 126 pages
File Size : 40,25 MB
Release : 2014-06-05
Category : Mathematics
ISBN : 0821898868
Book Description
The authors define combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an oriented -manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology. Their proof uses a formula for the Viterbo-Maslov index for a smooth lune in a -manifold.
A Combinatorial Proof of the Invariance of Tangle Floer Homology
Author : Timothy Adam Homan
Publisher :
Page : 124 pages
File Size : 15,85 MB
Release : 2019
Category : Electronic dissertations
ISBN :
Book Description
The aim of this work is to take the combinatorial construction put forward by Petkova and Vértesi for tangle Floer homology and show that many of the arguments that apply to grid diagrams for knots can be applied to grid diagrams for tangles. In particular, we showed that the stabilization and commutation arguments used in combinatorial knot Floer homology can be applied mutatis mutandis to combinatorial tangle Floer homology, giving us an equivalence of chain complexes (either exactly in the case of commutations or up to the size of the grid in stabilizations). We then added a new move, the stretch move, and showed that the same arguments which work for commutations work for this move as well. We then extended these arguments to the context of A-infinity structures. We developed for our stabilization arguments a new type of algebraic notation and used this notation to demonstrate and simplify useful algebraic results. These results were then applied to produce type D and type DA equivalences between grid complexes and their stabilized counterparts. For commutation moves we proceeded more directly, constructing the needed type D homomorphisms and homotopies as needed and then showing that these give us a type D equivalence between tangle grid diagrams and their commuted counterparts. We also showed that these arguments can also be applied to our new stretch move. Finally, we showed that these grid moves are sufficient to accomplish the planar tangle moves required to establish equivalence of the tangles themselves with the exception of one move.
Sergei Gukov, Mikhail Khovanov, and Johannes Walcher
Author : Sergei Gukov:
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 15,18 MB
Release : 2016-12-23
Category : Mathematics
ISBN : 1470414597
Book Description
Throughout recent history, the theory of knot invariants has been a fascinating melting pot of ideas and scientific cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory, and quantum gravity. The 2013 Séminaire de Mathématiques Supérieures in Montréal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology, from the mathematical background to the most recent developments, and provided an access point to the relevant parts of theoretical physics as well. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field.
Knots and Links
Author : Dale Rolfsen
Publisher : American Mathematical Soc.
Page : 458 pages
File Size : 12,69 MB
Release : 2003
Category : Mathematics
ISBN : 0821834363
Book Description
Rolfsen's beautiful book on knots and links can be read by anyone, from beginner to expert, who wants to learn about knot theory. Beginners find an inviting introduction to the elements of topology, emphasizing the tools needed for understanding knots, the fundamental group and van Kampen's theorem, for example, which are then applied to concrete problems, such as computing knot groups. For experts, Rolfsen explains advanced topics, such as the connections between knot theory and surgery and how they are useful to understanding three-manifolds. Besides providing a guide to understanding knot theory, the book offers 'practical' training. After reading it, you will be able to do many things: compute presentations of knot groups, Alexander polynomials, and other invariants; perform surgery on three-manifolds; and visualize knots and their complements.It is characterized by its hands-on approach and emphasis on a visual, geometric understanding. Rolfsen offers invaluable insight and strikes a perfect balance between giving technical details and offering informal explanations. The illustrations are superb, and a wealth of examples are included. Now back in print by the AMS, the book is still a standard reference in knot theory. It is written in a remarkable style that makes it useful for both beginners and researchers. Particularly noteworthy is the table of knots and links at the end. This volume is an excellent introduction to the topic and is suitable as a textbook for a course in knot theory or 3-manifolds. Other key books of interest on this topic available from the AMS are ""The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes"" and ""The Knot Book.""
