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

中文翻译：

---

Crosscap 结数和体积界限

在本文中，我们通过使用我们最近引入的图解结不变量（定理 1）来获得任何交替结的跨帽数。证明由和弦图的属性给出（Kindred 通过其他技术独立证明了定理 1）。对于非交替结，我们给出了定理 1 的推广定理 2。我们还改进了已知的公式来获得交叉帽数量的上限（交替或非交替）（定理 3）。作为推论，本文将交叉帽数和我们的不变量与其他结不变量联系起来，例如琼斯多项式、扭曲数、交叉数和双曲线体积（推论 1-7）。在附录 A 中，使用定理 1，我们完成了具有多达 11 个交叉点的交替结的跨帽数，包括以前未知值的 [公式：