Problems.
 What knots with high symmetries have projections that
demonstrate
this symmetry? (eg. the Figure8 knot)
 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.]
 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)
 Hass and Lagrias proved that if you have an ncrossing 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.
 Find a pair of nontricolorable 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.)
 If a knot is pcolorable, then for which q is that knot also
qcolorable?
(eg. It is true for q=kp, k an integer, by using the (pcoloring) x
k.)
 Is there a pair of mutant knots distinguished by tricolorability?
Answer: Colin Starr has a proof that the answer is no.
 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?
 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.
 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.
 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.)
 Is there a 2component link with trivial components so that in
order
to realize its splitting number, one must knot both components?
 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.)
 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.
 Investigate G_n(K) which is the least genus of a singular
npunctured
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.
 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.
