# How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$.

In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings of a knot diagram and determine whether it is a knot or a link?

I also have a few questions - I wasn't able to find information on the web but perhaps you could point me to closely related concepts:

• Is it possible for changing the direction of one or more crossings on a knot diagram (in $\mathbb{R}^2$) to result in a knot being changed to a link or vice versa?

• Do there exist links with canonical knot diagrams in $\mathbb{R}^2$ that are made up of more than two "components"?

Thanks!

• 1. Yes, but the problem is NP hard. 2. Impossible. 3. Yes, of course, just draw two disjoint circles in the plane. May 23, 2014 at 23:32
• I'm most confident that the problem of checking whether a diagram represents a knot or a link is not NP-hard. One just needs to travel along the diagram, and checking if during the journey one exhausts all crossings. This should be linear in the number of crossings. May 24, 2014 at 11:22