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Reviewed for Topology Atlas by Corinne Cerf
KNOT THEORY by K. REIDEMEISTER.
Originally published as KNOTENTHEORIE by K. REIDEMEISTER, Ergebnisse der Mathematik und ihrer Grenzgebiete, Alte Folge, Band 1, Heft 1, SPRINGER, Berlin, 1932.
Translated from the German and edited by L. F. BORON, C. O. CHRISTENSON, and B. A. SMITH, BSC ASSOCIATES, Moscow, Idaho, U.S.A., 1983.
This book is a 1983 translation of the 1932 celebrated book by Kurt Reidemeister. It is subdivided into three chapters.
The first one is an introduction to knots and braids, including (a sketch of) the original proof that two knots are equivalent if and only if their projections are related by a finite sequence of the three so-called Reidemeister moves.
The second chapter describes the main knot invariants obtainable from matrices, like linking numbers, torsion numbers, determinants, and L-polynomials, now called normalized Alexander polynomials, that have been discovered independently by Reidemeister and Alexander.
The third chapter deals with knot groups: definition by generators and relations from a projection, invariance, equivalence with the fundamental group of the knot complement, calculation of the group of special families of knots. A group-theoretic interpretation of the matrices and L-polynomials of Chapter II is given.
Although the quality of print and illustrations of this 1983 printing is not as good as that of modern books, it is impressive to see how its content remains up-to-date for the whole of knot theory preceding the "Jones revolution". Where Reidemeister's terminology was different from the 1983 terminology, the translators adapted it, while respecting the spirit of the original book. There still have been some minor terminology changes since then, but they do not hinder the intelligibility of the text.
This landmark in the history of knot theory should have a choice place on the shelves of all knot theorists.
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