# Crosscap numbers of alternating knots via unknotting splices

## TL;DR

This paper proves that the splice-unknotting number bounds the crosscap number exactly for prime alternating knots and computes these numbers for all such knots up to 13 crossings using Gauss codes.

## Contribution

It confirms that the splice-unknotting number equals the crosscap number for prime alternating knots and provides a method to compute these numbers for knots up to 13 crossings.

## Key findings

- The splice-unknotting number bounds the crosscap number exactly for prime alternating knots.
- Computed crosscap numbers for all prime alternating knots up to 13 crossings.
- Validated the conjecture for a broad class of knots using Gauss codes.

## Abstract

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.