Applications of Knot Theory by Dorothy Buck and Erica Flapan, Dorothy Buck, Erica Flapan

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

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

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THOM introduced the spaces above in order to convert D,. into a stable homotopy group. We give now a bare outline of this noted theorem. 1) Thom. +k(MSO(k)), and m.. +k(MO(k)). The proof is long [40, 22]. Herewe merely define the isomorphisms. Suppose that M" is a closed oriented n-manifold. Embed M" in SnH via the Whitney embedding theorem. Denote by ~ : A -+ M" the normal cell bundle to M" in 5"+k. Assuming Sn+k oriented, we may assume the tangent bundle p, to Sn+k oriented; moreover the tangent bundle -r to Mn is oriented.

1) this is a monomorphism with image H .. (X, A; Z) ® QllcH.. o is a permanent cycle, so is every element of H7J(X,A ;Z) ® DaC H7l(X,A; Dq)· Now H7l(X, A; Z) ® Da is a direct summand of E;,q. the other summand TM being isomorphic to Tor(H'J)-l (X, A; Z), Da)· Since Da has no odd 42 II. Computation of the bordism groups torsion, Tv,a consists of 2-torsion only. 1), d 2 carries E~,q onto an odd torsion group, and since T v. a) = 0. Thus d2 = 0. As we continue through the spectral sequence, it is seen to be trivial.

Any Stiefel-Whitney dass of the product M: X vm is of form W; = 1 ® v1 + terms involving Stiefel-Whitney classes of M:. From dimensional considerations we see that W;, ... W;k(f;n)*cn,i = lt cn, i ® V;, ••• V;k' Thus N N (W;, ... W;k(l;n)*cn,i, an X am) N = = (I* (cn,i), an) (v;, ... V;k' am) (v;, ... = i, the Whitney numbers of the product associated with c"·; all vanish. Now suppose there is an expression Em,i [M:-m, 1;] 2 [V:'J 2 = 0. All the Whitney numbers must vanish. We show inductively that [V:'J 2 = 0 for all m, i.

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