Preface

Geometry and topology are subjects generally considered to be "pure"

mathematics. Both originated in the effort to describe and quantize shape

and form in order to understand the "real" world. Both enjoy a robust and

sustained internal intellectual life, abstracted from the "reality" of their ori-

gins. Recently, some of the methods and results of geometry and topology

have found new utility in both wet-lab and theoretical science. Conversely,

science is influencing mathematics, from posing questions which call for the

construction of mathematical models to the importation of theoretical meth-

ods of attack on long-standing problems of mathematical interest.

A case in point is the subject of knot theory, which is utilized to a greater

or lesser degree in each of the six papers in this volume. Knot theory traces

its mathematical origins to the work of Gauss on computing inductance of

linked circular wires, and to the work of Kelvin and Tait on the vortex theory

of atoms. Knot theory is the study of entanglement and symmetry of elas-

tic graphs in 3-space. It has proven to be fundamental as a laboratory for

the development of algebraic topology invariants and in the understanding

of the topology of 3-manifolds. During the last decade, laboratory scientists

have become increasingly aware that the analytical techniques of geometry

and topology can be used in the interpretation and design of experiments.

Chemists have long been interested in developing techniques that will allow

them to synthesize molecules with interesting 3-dimensional structure (knots

and links). Polymer scientists study the chemical and physical ramifications

of random topological entanglement in large molecules. Models for molecu-

lar structure must be built and understood; reactions which produce specific

3-dimensional shapes must be designed; chemical proof of structure must be

produced, and these proofs often involve the use of topology to interpret data

such as NMR (Nuclear Magnetic Resonance) spectra. Molecular biologists

know that the spatial conformation of DNA and the proteins which act on

DNA is vital to their biological function; moreover, differential geometry and

knot theory can be used to describe and quantize the 3-dimensional struc-

ture of DNA and protein-DNA complexes. Biologists devise experiments on

circular DNA which elucidate 3-D molecular conformation (helical twist, su-

percoiling, etc.) and the action of various important life-sustaining enzymes

(topoisomerases and recombinases). These experiments are often performed

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