By Akio Kawauchi
Knot thought is a speedily constructing box of analysis with many functions, not just for arithmetic. the current quantity, written by means of a well known professional, offers a whole survey of this idea from its very beginnings to present day most up-to-date study effects. An integral e-book for everybody keen on knot theory.
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Additional resources for A Survey of Knot Theory
15 Let Si and Sk be two Seifert circles in the system of Seifert circles of a link diagram D. Assume that Sk is outside Si and Si is outside Sk and their orientations are coherent. 16. Then we deform D by Reidemeister moves as follows: At first, stretch out Sk n b along b until CHAPTER 1 PRESENTATIONS 16 it is near Si. 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.
3 Establish a similar classification for torus links. 4 Find all of the torus links with crossing number S 10. 3 Pretzel links For non-zero integers ql, q2, ... 1 is called the pretzel link and denoted by P(ql, q2, ... , qm), where qi indicates Iqil crossing points with sign E = qi/lqil = ±1. Suppose that (q~, q&, ... , q~) is a cyclic permutation of (ql, q2, ... , qm). Then P(q~, q&, ... , q~) and P(ql, q2, ... , qm) are positive-equivalent. If qi = ±1, then P(ql, ... , qi, ... , qrn) is positive-equivalent to P(qi, ql, ...
7) of any pretzel link is a reflection group in 2-dimensional spherical, Euclidean or hyperbolic space, which has been known since the appearance of [Reidemeister 1932]. Chapter 3 Compositions and decompositions In this chapter, we discuss how to construct a new link from given links by various compositions. Then we discuss decompositions, which are the inverse operations of compositions. After that, compositions of tangles are discussed. Throughout this chapter, links are understood to be links in 83.