A knot invariant which highlights the unknot

Question

Is there a knot (link) invariant, or a combination of them which discern the unknot (unlink) from any other knot (link)?

Answer 1

Yes, however they are not known to be computable in polynomial time. A quick an incomplete survey:

• The Seifert genus of a knot is $0$ if and only if it is unknotted.

• The fundamental group of the knot complement (the knot group) is isomorphic to $\mathbb{Z}$ if and only if the knot is unknotted.

• The A-polynomial (not to be confused with the Alexander polynomial), which has to do with homomorphisms of the knot group to $\mathrm{SL}_2\mathbb{C}$.

• Both knot Floer homology and Khovanov homology detect unknots.

• For knots with 10 or fewer crossings, the Alexander polynomial suffices.

The unknotting problem is known to be in NP and coNP. That means, given a knot presented in some manner, someone can give you a proof that the knot is either knotted or unknotted that takes polynomial time to verify.

Wikipedia has a list of unknotting algorithms. Approaches include:

• enumerating Seifert surfaces to find a disk that bounds the knot.
• exhaustively trying all sequences of Reidemeister moves of a diagram up to a certain polynomial number of steps.
• exhaustively trying all sequences of Pachner moves of a triangulation of the knot complement up to a certain polynomial number of steps, until it is obviously an unknot triangulation.
• use Dynnikov's result that arc presentations (grid diagrams) for an unknot can be monotonically simplified, so try all monotonic simplifications.

There are also a number of invariants where whether it can detect unknots is an open question, such as the Jones polynomial, the HOMFLY polynomial, and finite-type (Vassiliev) invariants.