Library of Congress Cataloging-in-PublicationData Murasugi, Kunio, 1929-[Musubime riron to sono 6y5. English] Knot theory and its applications / Kunio Murasugi ; translated by Bohdan Kurpita. p. cm. Includes bibliographical references (p. -) and index.

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INTRODUCTION AND IMAIN THEOREMSLETL bea tame link in S3 and VL(f) the Jones polynomial of L defined in [Z]. For a projection E of L, c(L) The primeness is necessary in the last statement ofTheorem B, since the connected sum of two figure eight knots is alternating, but it has a minimal non-alternating projection. Note that the figure eight knot is amphicheiral.Theorems A and B follow easily from Theorems l-4 (stated below) which show strong connections between c(L) and the Jones polynomial Vr(t). Let d maxVL(t) and d,i,v~(t) denote the maximal and minimal degrees of V,(t), respectively, and span V,(t) = d,,, Vr(t) -d,;,VL(t). If L is an alternating link, then we are able to prove the following:l The work was done while the author was visiting at the University of Geneva, Switzerland.

LETL bea tame link in S3 and VL(f) the Jones polynomial of L defined in [Z]. For a projection E of L, c(L) denotes the number of double points in L and c(L) the minimum number of double points among all projections of L. A link projection t is called proper if L does not contain "removable" double points-. like ,<_: or /.+_,. \/;'I In this paper, we will prove some of the outstanding classical conjectures due to P.G. Tait [7]. THEOREM A. (P. G. Tait Conjecture) Two (connected and proper) alternating projections of an alternating link have the same number of double points. THEOREM B. The minimal projection of an alternating link is alternating. In other words, an alternating link always has an alternating projection that has the minimum number of double points among all projections. Moreover, a non-alternating projection of a prime alternating link cannot be minimal.

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