Knot Theory Book: Intro for Topology & Algebraic Topology

Thread starter blerg
Start date Apr 12, 2011
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Book Knot theory Theory

In summary, Knot theory is a branch of mathematics that studies mathematical knots using topology and algebraic topology. It has numerous applications in fields such as biology, chemistry, physics, and computer science, and is connected to other areas of mathematics such as geometric group theory and low-dimensional topology. It differs from topology in that it specifically focuses on studying knots and their properties. Algebraic topology is used in knot theory to assign algebraic invariants to knots, and knot theory has real-life applications in DNA research, materials science, computer graphics, fluid dynamics, and cryptography.

Apr 12, 2011

blerg

I'm looking for an introductory book for knot theory. I have background in topology and algebraic topology. I would prefer a more sophisticated treatment, but I have no previous knowledge.

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Apr 15, 2011

xristy

Gold Member

Cromwell; "Knots and Links"; Cambridge, 2004, is thorough, sophisticated and up to date.

1. What is Knot Theory?

Knot theory is a branch of mathematics that studies mathematical knots, which are closed curves in three-dimensional space. It involves the use of topology and algebraic topology to classify and understand the properties of knots.

2. Why is Knot Theory important?

Knot theory has applications in various fields, such as biology, chemistry, physics, and computer science. It also has connections to other areas of mathematics, such as geometric group theory and low-dimensional topology. Additionally, it helps us understand the structure and behavior of complex systems.

3. What is the difference between Knot Theory and Topology?

Topology is the study of the properties of space that are preserved under continuous deformations, while Knot theory specifically focuses on studying knots and their properties. Knot theory is a subfield of topology and uses topological techniques to study knots.

4. How is Algebraic Topology related to Knot Theory?

Algebraic topology is a branch of mathematics that uses algebraic techniques to study topological spaces. In knot theory, algebraic topology is used to assign algebraic invariants to knots, which can then be used to classify and distinguish between different knots.

5. What are some real-life applications of Knot Theory?

Knot theory has been used in various fields, such as DNA research, materials science, and computer graphics. It has also been applied in the study of fluid dynamics and knotted proteins. Additionally, knot theory has applications in cryptography, where it is used to create secure codes and passwords.