A Survey of Knot Theory by Akio Kawauchi

By Akio Kawauchi

Knot conception is a quickly constructing box of study with many functions not just for arithmetic. the current quantity, written via a well known expert, supplies an entire survey of knot thought from its very beginnings to cutting-edge latest learn effects. the subjects comprise Alexander polynomials, Jones kind polynomials, and Vassiliev invariants. With its appendix containing many helpful tables and a longer checklist of references with over 3,500 entries it's an critical ebook for everybody fascinated about knot concept. The e-book can function an creation to the sector for complicated undergraduate and graduate scholars. additionally researchers operating in open air parts similar to theoretical physics or molecular biology will make the most of this thorough research that is complemented via many routines and examples.

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18. 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 is a trivial 2-string tangle. 3 is the 2-bridge knot C(an-l, a n -2, ... , a2, al) (cf. 1). The trivial 2-string tangle with Conway notation al a2 ... an is also called a rational tangle with slope 1 an + 1 ' an-l + ... +al CHAPTER 3 COMPOSITIONS AND DECOMPOSITIONS 36 which is a rational number or 00. 1a by ambient isotopies keeping the boundary fixed. Fig. 3 For two tangles (A, s) and (B, t), suppose that the numbers of points in and in are equal. p : a(B, t) ----+ a(A, s). 6. 4, the link type of a tangle sum is not uniquely determined by the tangles.

1 whose boundaries belong to the same trefoil knot type. Show that a is not orientable, thus it is not a Seifert surface, and that b is a Seifert surface which is ambient isotopic to c in R3. ~~ ~~ a b c Fig. 3 For any oriented link L in R 3 , there exists a Seifert surface for L. x -D Fig. 2 Proof (Seifert's algorithm). Let D be a diagram of L in the z = 0 plane R2 in R3. 2. Then D' has no crossings and hence is the boundary of a collection of oriented disks in R2. We deform these disks into mutually disjoint disks by slightly pushing their interiors into the upper 47 48 CHAPTER 4 SEIFERT SURFACES I: A TOPOLOGICAL APPROACH half space.

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