Concordance crosscap numbers of knots and the Alexander polynomial

Abstract

For a knot K K the concordance crosscap number, c ( K ) c(K) , is the minimum crosscap number among all knots concordant to K K . Building on work of G. Zhang, which studied the determinants of knots with c ( K ) > 2 c(K) > 2 , we apply the Alexander polynomial to construct new algebraic obstructions to c ( K ) > 2 c(K) > 2 . With the exception of low crossing number knots previously known to have c ( K ) > 2 c(K) > 2 , the obstruction applies to all but four prime knots of 11 or fewer crossings.