By Dorothy Buck and Erica Flapan, Dorothy Buck, Erica Flapan

Over the last 20-30 years, knot concept has rekindled its ancient ties with biology, chemistry, and physics as a way of constructing extra subtle descriptions of the entanglements and homes of normal phenomena--from strings to natural compounds to DNA. This quantity is predicated at the 2008 AMS brief direction, functions of Knot thought. the purpose of the quick direction and this quantity, whereas now not masking all elements of utilized knot concept, is to supply the reader with a mathematical appetizer, with a view to stimulate the mathematical urge for food for additional examine of this intriguing box. No past wisdom of topology, biology, chemistry, or physics is thought. specifically, the 1st 3 chapters of this quantity introduce the reader to knot conception (by Colin Adams), topological chirality and molecular symmetry (by Erica Flapan), and DNA topology (by Dorothy Buck). the second one 1/2 this quantity is concentrated on 3 specific functions of knot conception. Louis Kauffman discusses functions of knot concept to physics, Nadrian Seeman discusses how topology is utilized in DNA nanotechnology, and Jonathan Simon discusses the statistical and lively homes of knots and their relation to molecular biology.

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24) is satisfied. 3) The point zy is in s. 24), for a ·suitable choice of the constant () the surface Xo passes through the point M0 • On the other hand, it may be always assumed that the function f(z) is bo:unded in the domain D. In addition lwil < r 1 for all points of the set S. in common with the ·set S. Therefore, the family of surfaces {xe} must possess all the three properties stated in Lemma 5. This is impossible and thus Lemma 6 has been proved. n;, . ,, . COROLLARY 1. Let-the set 0' be decomposed into domains 0i, In each of these domains the functions R;(z 1) are logarithmically subharmonic.

V=O Here P 11 (z, f) are polynomials in zl' • • •, zn, fi (z), • • •, fN (z) ). 27) {,y (z). PROOF. +i:iv of the variables Zp ••• ' Zn' wl, ••• 'WN consider the set :E* = {z E~*. w 1 = f/z), j = 1, ... , N). By the preceding lemma, 7Tl (~*) = ~*. Therefore there exists a neighborhood ~* € rr 0 such that in its connected components ~ ::> /1 the function F(z) remains holomorphic. 27). REMARK. (z) belong to some complete family of functions I *, then, as is seen immediately from the definition· of the complete family, each §z. *

The Weil I) See, for example, I. I. Privalov, Introduction to the theory of functions of a com plex variable, Gostehizdat, Moscow-; I954, p. 420 (Russian). , §2. / (z) 41 are to be re- placed by polynomials. en -1 PROOF. We first suppose that the lemma is true for the space show that then it is also true for the space en. and Next we consider the case n = 1. 77 1 (I,)= S and hence the set s are bounded. If the set O' coincides wit~ the whole zl -plane, then the assertion of our lemma for the space is evident The set en from the Corollaries 1 and 2 of Lemma 6.