Crosscap number of a knot

Hitoshi Murakami and Akira Yasuhara

Vol. 171 (1995), No. 1, 261–273

DOI: 10.2140/pjm.1995.171.261

Abstract

B. E. Clark 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.

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 22 October 1992

Revised: 8 February 1993

Published: 1 November 1995