Open Questions

Colin Adams

Hyperbolic knots

1. Prove that there are no closed embedded totally geodesic surfaces in the complements of hyperbolic knots.
2. Find additional examples (beside the three known) of hyperbolic knots that have closed immersed totally geodesic surfaces in their complements.
3. Meridian length in a maximal cusp of a knot is known to be between 1 and 6, with 1 realized only for the figure-eight knot. Examples are known with meridian length approaching 4 from below. Prove that meridians have length strictly less than 4.
4. Find an example of a knot with nonorientable totally geodesic Seifert surface.
5. Prove that every closed orientable 3-manifold contains a knot with a hyperbolic complement containing a totally geodesic surface.
6. Prove that unknotting tunnels for hyperbolic knots of unknotting number 1 are realized by geodesics.

Knots in general

1. The stick number of a knot is the least number of sticks glued end-to-end to construct the knot. Does the stick number of K equal the equal-length stick number of K? (Candidate for a counterexample is 8₁₉).
2. Suppose that K is made from n equal length sticks. Can you construct K from n+1 equal length sticks? (Note that you can construct it from n+2 equal length sticks. Also, if there were an "outermost stick", you would be able to do so.)
3. Does there exist a pair of mutant knots with different stick number?
4. Does there exist a knot such that any one crossing change in any minimal crossing projection of the knot yields knots of the same crossing number? (Note that there cannot be any bigons in any minimal crossing projection.)
5. Two knots are 3-equivalent if we can obtain one from the other by a sequence of ±3-moves. Show that every knot is 3-equivalent to either the trivial knot or a trivial link. Maybe show this for certain classes of knots.