By Akio Kawauchi

ISBN-10: 3034892276

ISBN-13: 9783034892278

ISBN-10: 303489953X

ISBN-13: 9783034899536

Knot concept is a swiftly constructing box of analysis with many functions not just for arithmetic. the current quantity, written via a widely known expert, provides a whole survey of knot idea from its very beginnings to cutting-edge newest study effects. the themes comprise Alexander polynomials, Jones kind polynomials, and Vassiliev invariants. With its appendix containing many helpful tables and a longer record of references with over 3,500 entries it really is an fundamental booklet for everybody enthusiastic about knot thought. The booklet can function an creation to the sector for complicated undergraduate and graduate scholars. additionally researchers operating in outdoor parts reminiscent of theoretical physics or molecular biology will make the most of this thorough learn that's complemented via many routines and examples.

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**Additional resources for A Survey of Knot Theory**

**Example text**

J5 3 4 b a Fig. 2 Prove that S( a, (3) is invertible. 3 (1) The 2-bridge knots S(a,j3) and S(a',j3') belong to the same type if and only if a = a', j3±1 == j3' (mod oJ (2) The 2-component 2-bridge links S(a, (3) and S(o/, j3') belong to the same type if and only if a = a', j3±1 == j3' (mod 2a). If we consider positive-equivalence instead of the link type, then the condition of (2) reduces to that of (1). Proof. When we ignore the orientation, the proof follows from the classification of the lens space (cf.

We say that two braids are strongly equivalent if there is an ambient isotopy as above with the extra condition that for each level t, it (b o ) is a braid. These two equivalence relations on braids are actually the same equivalence relation ([Artin 1947]). Let b1 C Ir and b2 c Ii be two n-string braids. We construct a new braid b1 b2 in the cube Ir U Ii by attaching the bottom face of Ir to the top face of Ii naturally (see figure 1. 1). ) This braid b1 b2 is called the product of b1 and b2 . The quotient space of the set of n-string braids modulo the equivalence relation above becomes a group with this product operation.

2 Show that a link L is a b-bridge link if and only if L has a 2b-plat presentation. SUPPLEMENTARY NOTES FOR CHAPTER 1 19 Supplementary notes for Chapter 1 Alexander proved that any link type can be presented by a closed braid in [Alexander 1923]. 2 is due to [Yamada 1987]. Usually, the linking number is defined for simplicial cycles by using the intersection numbers of simplicial chains. The intersection number of simplicial chains and the linking number for simplicial cycles are described in [Seifert-Threlfall 1980].

### A Survey of Knot Theory by Akio Kawauchi

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