Narrow Results

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two.

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for $193$ knots (Tables A.1).

We use the methods of Hedden, Juhász and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_n which will bound at least 2^2n-1 non-isotopic minimal spanning surfaces of genus n.

We give a lower bound of the crosscap number of alternating knots using band surgery. This lower bound is equal to the minimum number of the first Betti number, i.e. one minus the maximal Euler characteristic, over all (orientable and non-orientable) surfaces spanning a given alternating knot.

We have listed all families of alternating knots with the crosscap number three.

Let L be an unsplittable link in S³ and let V denote the double cover of S³ branched along L. Starting with an arbitrary planar projection of L, it is shown how to construct a special presentation for the fundamental group π₁V which encodes in a simple way geometric data from a (checkerboard) spanning surface obtained from the given projection, and which relates to the construction of Coxeter quotients of π₁ (S³-L) previously studied by the author and Y. W. Lee.