Unknotting number formally?

Question

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings $S^1\hookrightarrow\mathbb{R}^3$ required besides continuity for unknotting number to exist?) What is a reference for a formal proof that every knot in some category (maybe PL?) has finite unknotting number?

Answer 1

A (say PL) knot diagram $D$ can be thought of as a (PL) immersed curve in the plane with only transverse double points such that each of the double points carries "crossing information", i.e. it is decorated in a way to depict which strand passes over the other. A crossing change just alters the decoration at one of the double points so that the over and under strands are interchange.

The unknotting number $u(D)$ of a diagram $D$ is the minimum number of crossing changes necessary to change $D$ into a diagram of the unknot. The unknotting number of a diagram is always finite, and here's one way to see that. Let $p$ be some basepoint on the knot away from the crossings. Start traveling along the knot from $p$ (in an arbitrary direction). Perform crossing changes (if necessary) so that the first time you encounter any crossing, you encounter it along an "over" strand. Such a diagram is always the unknot. Suppose that the diagram is in the $(x,y)$-plane, and that direction of projection is the $z$-axis. Then the embedding of the knot in $\mathbb{R}^3$ can be taken to be decreasing in the $z$-direction except for one line segment that parallel to the $z$-axis that projects to the point $p$.

The unknotting number $u(K)$ of the knot $K$ is then defined as $$u(K) = \min\{u(D)~|~D~\text{is a diagram of }K\}.$$ Since each $u(D)$ is finite, of course $u(K)$ is also finite.