Lecture Notes on Knot Invariants

The volume is focused on the basic calculation skills of various knot invariants defined from topology and geometry. It presents the detailed Hecke algebra and braid representation to illustrate the original Jones polynomial (rather than the algebraic formal definition many other books and research articles use) and provides self-contained proofs of the Tait conjecture (one of the big achievements from the Jones invariant). It also presents explicit computations to the Casson–Lin invariant via braid representations. With the approach of an explicit computational point of view on knot invariants, this user-friendly volume will benefit readers to easily understand low-dimensional topology from examples and computations, rather than only knowing terminologies and theorems. Contents:Basic Knots, Links and Their EquivalencesBraids and LinksKnot and Link InvariantsJones PolynomialsCasson Type Invariants Readership: Undergraduate and graduate students interested in learning topology and low dimensional topology. Key Features:Applies a computational approach to understand knot invariants with geometric meaningsProvides a complete proof of Tait's conjectures from an original Jones polynomial definitionGives recent new knot invariants from the approach of algebraic geometry (characteristic variety)Readers will get a hands-on approach to the topological concepts and various invariant, instead of just knowing more fancy wordsKeywords:Knot Classifications;Tait Conjectures;Reidemeister Moves;Characterization of Braid Representation;Unknotting Number;Bridge Number;Linking Number;Crossing Number;Wirtinger Presentation;Magnus Representation;Twisted Alexander Polynomial;Hecke Algebra;Ocneanu Trace;Jones Polynomial;Kauffman Bracket;Casson Type Invariant