Alternating and Non-Altenating Knot projections with same crossing number?

Question

The usual way to represent the alternating prime knots is via a reduced, alternating projection which of course shows the minimal crossing number for that knot. But could there be a similar non-alternating projection for one of these knots?

That is to say: Is there an alternating knot which has a non-alternating projection which shows that knot's minimal crossing number?

Answer 1

The answer to your question is no, that is, every minimal crossing diagram of a prime alternating knot (or link) is alternating. Here's why.

Let $D$ be a prime knot diagram, and let $c(D)$ be the number of crossings in that diagram. Kauffman, Murasugi, and Thistlethwaite showed that the span of the Kauffman bracket of $D$ gives a lower bound on $c(D)$: $$\operatorname{span}\langle D \rangle \leq 4 c(D).$$ Moreover, they showed that this inequality is an equality if and only if $D$ is a reduced alternating diagram. Since $\operatorname{span}\langle D \rangle$ is a knot invariant, we have the desired result.

Details of these proofs can be found in many knot theory text books, or in the original papers, or in the related paper of Turaev:

1. Kauffman. State models and the Jones polynomial.
2. Murasugi. Jones polynomials and classical conjectures in knot theory.
3. Thistlethwaite. A spanning tree expansion of the Jones polynomial.
4. Turaev. A simple proof of the Murasugi and Kauffman theorems on alternating links. (I could not find an online version of this paper).

Edit: In the composite case, there are minimal crossing diagrams that are non-alternating. The square knot, pictured below, is an alternating knot, and its minimal crossing diagram has six crossings. The picture below is a non-alternating diagram of the square knot also with six crossings.