A List of Approachable Open Problems in Knot Theory

(.ps and .pdf files also available)

Suggested by Colin Adams during the Knot Theory Workshop at Wake Forest University during June 24-28, 2002.

Problems.

  1. What knots with high symmetries have projections that demonstrate this symmetry? (eg. the Figure-8 knot)
  2. Find specific families of knots satisfying the property c(K_1#K_2) = c(K_1)+c(K_2), where c=c(K) is the crossing number and # means knot composition. (eg. This is known for alternating knots.) What about torus knots? [In a 2003 preprint, Yuanan Diao demonstrated that this does hold for compositions of torus knots, as well. This was also independently proved by Herman Gruber. His paper is available at arxiv.org under math.GT/0303273.]
  3. When is a knot equivalent to its inverse? (The inverse has the same projection but with an opposite orientation). (eg. the trefoil and its inverse)
  4. Hass and Lagrias proved that if you have an n-crossing projection of the trivial knot, you can turn it into a trivial projection by using no more than 2^(1,000,000,000n) Reidemeister moves. Find a better upper bound.
  5. Find a pair of non-tricolorable knots whose composition istricolorable or show that this is not possible. (To show it's false, it's enough to show that an open knot is tricolorable if and only if its closure is tricolorable.)
  6. If a knot is p-colorable, then for which q is that knot also q-colorable? (eg. It is true for q=kp, k an integer, by using the (p-coloring) x k.)
  7. Is there a pair of mutant knots distinguished by tricolorability? Answer: Colin Starr has a proof that the answer is no.
  8. What knots have the property that in their minimal crossing projection changing any 1 (or, in general, k) crossing(s) transforms it to the trivial knot?
  9. Find the simplest (i.e. fewest number of crossings) projection of the trivial knot that requires an increase in number of crossings to get to the minimal projection of the trivial knot through Reidemeister moves.
  10. Are there round Brunnian links with 3 or more components? (A link is round if it can be realized by components, each of which is a perfect circle.) Are there round Brunnian links with 3 or more components? (A link is round if it can be realized by components, each of which is a perfect circle.) [Michael Freedman has proved the answer is no. See Ian Agol's nice description of the proof at http://www.math.uic.edu/~ago1/circles.pdf.
  11. Is the trefoil the only nontritangent knot? (A knot is nontritangent if there is a realization of that knot that does not have any planes tangent to the knot at three or more points.)
  12. Is there a 2-component link with trivial components so that in order to realize its splitting number, one must knot both components?
  13. Find families of knots for which |g(K)-g_N(K)| -> infinity and g_N(K) > g(K) [or g_N(K) < g(K)]. (g(K) is the least genus of an orientable surface with boundary the knot. g_N(K) is the least genus of a nonorientable surface with boundary the knot.)
  14. Prove or disprove the following equality: G_0(K) = min{g(K), g_N(K)}. Here, G_0(K) is the least genus of any unpunctured singular surface with boundary the knot.
  15. Investigate G_n(K) which is the least genus of a singular n-punctured surface bounded by the knot. Note that if n= c, where c is the crossing number, then, G_n(K) = 0, by taking the cone of the projection to a point.
  16. It might be true that any alt. graph with no trivalent vertices has minimal crossing number in any reduced alt. projection with no uncrossed cycles.