Crosscap number of knots and volume bounds

Abstract

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 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 [Formula: see text] knots (Tables A.1).

Crosscap numbers of alternating knots via unknotting splices

“…Ito-Takimura define u − (D) to be the minimum number of u − splices among these splice-unknotting sequences. 10 They prove:…”

“…[1, 4, 7], [2, 1, 3], [5,4,2,3,6], [5,6,7] Crossings around B-faces: [4,5,7], [5,6], [6,3,1,7] Edges around A-faces: [14,4,10], [8, 1, 3], [12,5,9,2,7], [13,6,11] Edges around B-faces: [4, 1, 9], [2, 8], [10,5,13], [6,12], [11,7,3,14] D G [n] and find a Gauss code D G [n][K] for a reduced alternating diagram of K . Then we clean up this data by replacing each Gauss code with its reduced form.…”

Slope norm and an algorithm to compute the crosscap number

“…Among these is a formula for the crosscap number of torus knots by Teragaito [22]; an algorithm for alternating knots developed by Adams and Kindred [1]; or upper and lower bounds for the general case via the Jones polynomial by Kalfagianni and Lee [17]. Recent work by Ito and Takimura [8,9,10] establishes various further bounds. The KnotInfo data base [18], and in particular their page on crosscap numbers, gives a detailed overview and results for specific knots.…”

Plumbing and computation of crosscap number

“…The recent paper [4] introduces an unknotting-type number u − (K) of a knot K. It is known that u − (K) equals the crosscap number C(K) for every prime alternating knot K [6,5,8] 1 . The following question arises.…”