A Survey of Knot Theory by Akio Kawauchi

By Akio Kawauchi

Knot idea is a quickly constructing box of analysis with many purposes not just for arithmetic. the current quantity, written via a widely known professional, supplies an entire survey of knot conception from its very beginnings to modern latest examine effects. the subjects comprise Alexander polynomials, Jones sort 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 necessary e-book for everybody interested by knot concept. The ebook can function an advent to the sector for complicated undergraduate and graduate scholars. additionally researchers operating in outdoors components corresponding to theoretical physics or molecular biology will take advantage of this thorough learn that is complemented by means of many workouts and examples.

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Another proof was given in [Burde 1975] by using the linking numbers of branched covering spaces. Conway's normal form was re-introduced in [Conway 1970] through the tangle theory SUPPLEMENTARY NOTES FOR CHAPTER 2 29 (cf. Chapter 3). See also [Siebenmann *] on this matter. 14, see [Kanenobu-Miyazawa 1992]. [Schreier 1924]). Now the Alexander polynomial is an easy tool for solving this problem (cf. 4). The torus knots are characterized as the only knots whose groups have non-trivial centers (cf.

This deformation of the system of Seifert circles is called a concentric deformation of type II. If we consider the system of Seifert circles on the sphere, the concentric deformation of type II is nothing but the concentric deformation of type I. We note that a concentric deformation may increase the number of connecting arcs, but never changes the number of Seifert circles. Here we give an answer to the question mentioned before. Fig. 16 Fig. 17 Fig. 2 Any link diagram can be deformed into a braid presentation by a finite sequence of concentric deformations of types I and II.

2 are non-split. Here is a method for constructing a non-split tangle. 6 Let (e, v) be a tangle and D be a disk properly embedded in e such that D divides (e, v) into two tangles (A, s) and (B, t). We assume the following: (1) The numbers of points in (aA-D) nv, (aB -D) nv and Dnv are all greater than or equal to one. (2) Any disk ~ properly embedded in A with ~ n aD = 0 and ~ n s = 0 does not split s in A. (3) (B, t) is non-split. Then (e, v) is a non-split tangle. 6 above, we assume the following condi tion instead of condition (2): (2') The tangle (A, s) is non-split.

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