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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!

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  • $\begingroup$ 1. Yes, but the problem is NP hard. 2. Impossible. 3. Yes, of course, just draw two disjoint circles in the plane. $\endgroup$ May 23, 2014 at 23:32
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    $\begingroup$ 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. $\endgroup$ May 24, 2014 at 11:22

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About your second question: The correct formulation is "changing the sign" of the intersection, not changing the direction. One can only change the direction of an oriented segment, but clearly this is not what you meant, as one can't change the orientation of just one segment in a diagram if he wants the orientation of the link to stay well-defined. He would then have to change the orientation of every single segment belonging to the same connected component.

Now, changing the sign of an intersection boils down to switching between the overpassing and underpassing segments at the intersection point, so that the previously overpassing segment now becomes an underpass, and vice versa. But that clearly has nothing to do with the number of connected components of the link.

to avoid confusion, when I say "connected component" I mean connectedness in the usual topological meaning, and not whether said component is linked to another. As a counter-example for the latter, consider the Hopf link, where changing the sign of either intersection results in two unlinked trivial knots.

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