Download A Survey of Knot Theory by Akio Kawauchi PDF

Algebraic Geometry

By Akio Kawauchi

ISBN-10: 303489953X

ISBN-13: 9783034899536

Knot concept is a swiftly constructing box of study with many functions not just for arithmetic. the current quantity, written through a well known professional, offers an entire survey of knot idea from its very beginnings to modern day most modern study effects. the subjects contain Alexander polynomials, Jones variety polynomials, and Vassiliev invariants. With its appendix containing many helpful tables and a longer checklist of references with over 3,500 entries it truly is an integral ebook for everybody excited about knot thought. The publication can function an advent to the sector for complex undergraduate and graduate scholars. additionally researchers operating in outdoors components reminiscent of theoretical physics or molecular biology will reap the benefits of this thorough research that is complemented through many routines and examples.

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Example text

Further, the decomposition is unique in the following sense: if L1~L2~' .. ~Lm ~ L~~L~~· .. ~L~ for prime links Li(i = 1,2, ... , m) and Lj(j = 1,2, ... , n), then we have m = nand Li ~ L~ (i = 1,2, ... , n) after permuting the indices suitably. The proof is given by cut-and-paste argument in [Schubert 1949] and [Hashizume 1958]. 7 A link L is atoroidal if any torus T in the interior of the link exterior E(L) = c1(S3 - N(L)) of L is compressible or a-parallel in E(L) (see the terminology of Appendix C).

3 is a trivial knot, which is a contradiction. Fig. 2, a link obtained from two locally trivial tangles by a tangle sum need not be locally trivial. 1a. a c b d Fig. 5. 3d arc indivisible. 3b. 3d, for any dividing disk 40 CHAPTER 3 COMPOSITIONS AND DECOMPOSITIONS D, we consider two tangles obtained by dividing by D. 3c, one of them is trivial, since the arc in the original 3-ball is trivial. 1a. 6 A tangle is prime if it is non-split, locally trivial, and indivisible and if it is not a trivial I-string tangle.

Pn), with b the number of E. If b > 0 and Pi = -2E, then P(-Eb;PI,P2, ... ,Pn) has the same type as P(-E(b-1);PI,P2, ... ,-Pi, ... ,Pn)' Thus we can assume that none of Pi is equal to -2E when b > O. " ,Pn) to be a knot is that either n 2: 0 and all of the Pi'S and n + b are odd or that n 2: 1 and just one of the Pi'S is even. We 28 CHAPTER 2 STANDARD EXAMPLES call it a pretzel knot of odd type in the former case, and a pretzel knot of even type in the latter case. The pretzel knot is oriented by the orientation of the top arc running from right to left.

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