By Akio Kawauchi

Knot concept is a quickly constructing box of analysis with many purposes not just for arithmetic. the current quantity, written by means of a well known expert, provides a whole survey of knot thought from its very beginnings to latest most up-to-date study effects. the themes contain Alexander polynomials, Jones variety polynomials, and Vassiliev invariants. With its appendix containing many beneficial tables and a longer record of references with over 3,500 entries it really is an fundamental publication for everybody considering knot idea. The booklet can function an creation to the sector for complicated undergraduate and graduate scholars. additionally researchers operating in outdoors components akin to theoretical physics or molecular biology will take advantage of this thorough learn that is complemented through many workouts and examples.

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Let p be a self-map of (5, L n 5) of period 2 such that the fixed point set of p on 5 is a two-point set disjoint from L n 5, which we call a symmetry on (5, L n 5). There are three kinds of symmetries on (5,Ln5). 1 A link L' is an elementary Conway mutant of a link L if (53, L') is obtained from (53, L) by splitting along a Conway sphere 5 and re-gluing by using a symmetry p on (5, L n 5). A link L' is a Conway mutant of a link L if L' is obtained from L by a finite sequence of elementary Conway mutants.

3) A link has only trivial companions. [Hint: In the case of knots, (1) and (3) are equivalent. 3 Definition of a tangle and examples In this section, the concept of a tangle is introduced. 1 A tangle is the pair consisting of a 3-ball B3 and a (possibly disconnected) proper 1-submanifold t with at i= 0. In particular, it is an n-string tangle if t consists of narcs. Note that we do not consider the case at = 0 as a tangle (B, t). 2 A trivial (n-string) tangle is a tangle homeomorphic to the pair (D2, {aI, a2, .

We say that two elements of Bare Markov equivalent if they can be deformed into each other by a finite sequence of Markov moves. Then we have the following theorem: Fig. 21 Fig. 5 For two braids (b, n) and (b', n'), the vertically closed braids b and b' belong to the same link type if and only if (b, n) and (b', n') are Markov equivalent. See [Birman 1974] for the proof of this theorem. 2, it may be said that knot theory is the study of the Markov equivalence classes of the braid groups. , there is an algorithm to determine whether or not two given words are the same element in the braid group.