Marc Lackenby has several state-of-the-art theorems which shows a disconnect between studying knots via planar diagrams and studying their complement.
One such theorem states if you take n knots, K_1, K_2, …, K_n, and their connect-sum K = K_1 # K_2 # … # K_n, and if we let c(K) be the minimial number of crossings in a planar diagram for the knot K, we have the relationship:
(c(K_1)+c(K_2)+…+c(K_n))/152 <= c(K) <= c(K_1)+c(K_2)+…+c(K_n)
The 152 on the left-hand side is what gets my attention. Why isn’t it simply 1?
In a best-possible theorem of this type, I think the 152 *should* be a 1. That is, a minimal crossing diagram for a connect-sum of knots will simply be a planar connect-sum of the minimal crossing diagrams for the prime summands. Does anyone know an example where this isn’t the case?
Similarly, Lackenby has a result that describes a lower-bound on the number of crossings in a planar diagram for a satellite knot.
The above picture is a planar diagram for the Whitehead double of the figure-8 knot. The figure-8 knot has minimal crossing number 4 in a planar diagram. And I would like to say the above picture is a minimal-crossing number picture of the Whitehead double, which has 18 crossings. Lackenby’s theorem says the above knot must minimally have 4/(10^13) crossings in it! So clearly it’s not best-possible as there are no 1-crossing number knots. What I want to know is, what do people think is true? Do you think there is a simple prescription for the minimal crossing number of a connect sum, or perhaps even a satellite in general, and what do you think it should be?
There’s a rather direct upper bound on the minimal crossing-number of a satellite knot. If K is a satellite of a knot J using pattern L (meaning L=L_0 \cup L_1 is a 2-component link with the L_1 component the unknot). You can find a planar diagram for the link L so that L_1 is a round planar circle. Among such diagrams, you can minimize the number of crossings. Let “n” be the number of crossings in L_0’s diagram, and “m” the number of times L_0 passes through L_1’s planar disc. Let w(J) be the writhe of J, that is, the difference between the knot J’s homological framing and its planar framing. Then
c(K) <= n + (m^2)c(J) + m(m-1)|w(J)|
I imagine this formula must be in Schubert's Knotten und Vollringe but I haven’t looked at it in a while. The formula is mostly self-explanitory. The term “n” is the crossings of L_0 before you perform the satellite operation (since they’re still present in the planar diagram after you perform it). (m^2)c(J) are all the crossings in the satellite you get from the original crossings in J. m(m-1)|w(J)| are the new crossings you get because when you perform the satellite operation, you twist the strands. This term does not appear in the above diagram since the figure-8 knot has no writhe. The standard diagram of the trefoil has non-zero writhe.
The above formula, would it be too optimistic to conjecture that is an equality for all simple satellite knots? Does anyone have any ideas for a possible counter-example? By “simple” I mean in the JSJ-decomposition if one torus contains the other, that makes a partial order, and to be simple it would have to be a linear order. There are analogous formulas for non-simple JSJ-decompositions.
And how about for links? One could make a similar conjecture about their minimal crossing numbers. For example, with the link below.
This is a rather interesting 2-component link. It’s Brunnian — both components are trivial knots. This diagram has 40 crossings in it. I doubt there is a diagram with less than 40 crossings for this link. Presumably we’re nowhere close to knowing a proof of this. And I think this is a case where intuition fails — there are many isotopies of this link into other “nice” positions, how can you be sure this one has the least number of crossings?