1
$\begingroup$

Reidemeister's Theorem states:

Two knot projections $K_{1}$ and $K_{2}$ are equivalent if and only if $K_{2}$ can be obtained from $K_{1}$ by a sequence of Reidemeister moves.

Does this apply to links as well? If so, why? Is there a proof?

Clarification:

I'm not looking for a detailed topological proof - just a brief 'explanation' for beginners in Knot Theory without a background in topology.

Thanks!

$\endgroup$
2
  • $\begingroup$ Did you try to follow the same proof as for the knots? (I do not see any difference.) Check Rolphsen's book "Knots and Links". $\endgroup$ Mar 20 '14 at 10:40
  • $\begingroup$ @studiosus Sorry I should have clarified in my question. I haven't followed the proof for knots, I'm a beginner in Knot Theory and don't have a background in topology. Is there a kind of simple 'explanation'? $\endgroup$ Mar 20 '14 at 20:41
2
$\begingroup$

Yes, it does indeed apply to links. Informally you can think of Reidemeister moves as simple manipulations of an actual cord. If you make a knot out of actual physical cord and are able to manipulate it (without breaking/cutting the cord) into another knot, then of course they are equivalent and while you may not have noticed you did indeed only apply the Reidemeister moves. In the exact same way this applies to links.

Being as informal as possible:

In one direction say you have a "physical cord analog" of a link diagram $D_{1}$ and by whatever mindless jimble jamblings you get a cord analog of a link diagram $D_{2}$, then certainly these two diagrams of of the same link.

Then consider a situation where someone gives you two different pictures of diagrams $D_{1}$ and $D_{2}$ and tells you that they are equivalent. What if it was impossible to manipulate a physical analog of $D_{1}$ into an analog of $D_{2}$ (it may be by physical limitations, but assume you have some fantastic cord)? If it is impossible then $D_{1}$ can't be smoothly deformed into $D_{2}$, but this is precisely what it means for two diagrams to be of the same link! It then must be true that if $D_{1}$ and $D_{2}$ are two different diagrams of the same link then there must be a sequence of Reidemeister moves to get from one to the other.

$\endgroup$
1
$\begingroup$

Yes, Reidemeister's Theorem works for links. In his theorem, he proves that every ambient isotopy of a knot can be realized as a sequence of his three moves. Since knots are equivalent if there is an ambient isotopy between them, so working with diagrams, we use his moves instead. Similarly for links.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.