Crosscap number of a knot

Abstract

defined the crosscap number of a knot to be the minimum number of the first Betti numbers of non-orientable surfaces bounding it.In this paper, we investigate the crosscap numbers of knots.We show that the crosscap number of 74 is equal to 3.This gives an affirmative answer to a question given by Clark.In general, the crosscap number is not additive under the connected sum.We give a necessary and sufficient condition for the crosscap number to be additive under the connected sum. Introduction.We study knots in the 3-sphere S 3 .The genus g{K) of a knot K is the minimum number of the genera of Seifert surfaces for it [11].Here a Seifert surface means a connected, orientable surface with boundary K.In 1978, B. E. Clark [3] defined the crosscap number C(K) oϊK to be the minimum number of the first Betti numbers of connected, non-orientable surfaces bounding it.(For the trivial knot, it is defined to be 0 instead of 1.)He proved the following inequality and asked whether there exist knots for which the equality holds.C{K) < 2g(K) + 1.Note that since C (trivial knot) = 0, the equality does not hold for the trivial knot.In this paper, we give an example which satisfies the equality showing that C(7 4 ) = 3. (We use the notation of J. W. Alexander and B. G. Briggs for knots [1].See also [9] and [2].)Clark also studied how the crosscap number behaves under the connected sum.If we denote by T(K) the minimum number of the first Betti numbers of connected, unoriented surfaces bounding it (an unoriented surface means a surface which is orientable or not), T(K) is additive under the connected sum, i.e., Y(K^K 2 ) = Y(K λ ) + T(K 2 ) [3, Lemma 2.7] as H. Schubert proved for the genus [10].Note that Y(K) = min(2g(K), C(K)).A proof is given by an ordinary "cut-and-paste" argument.See for example [9, Theorem 5A14].But such an argument does not apply to the crosscap number because one of the two surfaces obtained from a non-orientable surface by cutting along an

Locations

• Pacific Journal of Mathematics - View - PDF