# Does Reidemeister's Theorem apply to Links?

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!

• 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". Mar 20 '14 at 10:40
• @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'? Mar 20 '14 at 20:41

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.