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Knot Theory

by K. Reidemeister

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.

Table of Contents


  1. Foreword to the English edition
  2. Publisher's foreword to the original edition
  3. Introduction
  4. Chapter I: -- Knots and their projections
    1. Definition of a knot
    2. Regular projections
    3. The operations $\Omega. 1, 2, 3$
    4. The subdivision of the projection plane into regions
    5. Normal knot projections
    6. Braids
    7. Knots and braids
    8. Parallel knots, Cable knots
  5. Chapter II: -- Knots and matrices
    1. Elementary invariants
    2. The matrices $(c^h_{\alpha \beta}}$
    3. The matrix $(a_{ik})$
    4. The determinant of a knot
    5. The invariance of the trosion numbers
    6. The torsion numbers of particular knots
    7. The quadratic form of a knot
    8. Minkowski's units
    9. Minkowski's units for particular knots
    10. A determinant inequality
    11. Classification of alternating knots
    12. Almost alternating knots
    13. Almost alternating circles
    14. The L-polynomial of a knot
    15. L-polynomials of particular knots
  6. Chapter III: -- Knots and Groups
    1. Equivalence of braids
    2. The braid group
    3. Definition of the group of a knot
    4. Invariance of the knot group
    5. The group of the inverse knot and of the mirror image knot
    6. The matrix $(l_{ik}(x))$ and the group
    7. The knot group and the matrices $(c^h_{\alpha \beta})$
    8. The edge path group of a knot
    9. Structure of the edge path group
    10. Covering spaces of the complementary space of the knot
    11. The group of a parallel knot
    12. The groups of torus knots
    13. The L-polynomials of parallel knots
    14. Several special knot groups
    15. A particular covering space
  7. Table of knots
  8. Bibliography
  9. Index

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