Low Dimensional Topology

March 25, 2012

Crossing numbers of knots

Filed under: 3-manifolds,Knot theory — Ryan Budney @ 4:05 pm

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?


  1. Additivity of the crossing number remains an outstanding open problem. Eg, http://jtopol.oxfordjournals.org/content/2/4/747.short

    Comment by Leonid Kovalev — March 25, 2012 @ 4:10 pm | Reply

  2. Correct Leonid, that’s basically the point of my post. But I wanted to say a little more than that. In particular, additivity of connect sum is an idea that fits into a theme. I wanted to expand on that theme with a more ambitious formula, for arbitrary satellites. If people know counter-examples to the more ambitious formula, perhaps that would lead to an insight into why additivity might not be true.

    Comment by Ryan Budney — March 25, 2012 @ 4:19 pm | Reply

  3. The arc index -2 is additive under connect sum: http://www.ams.org/mathscinet-getitem?mr=1339757

    I think one could probably use the techniques of Dynnikov to prove that the arc index of a satellite of K is greater than the arc index of K, with equality if and only if it is a trivial satellite. http://www.ams.org/mathscinet-getitem?mr=2232855

    Note that the arc index is the same as the minimal grid presentation, made popular by Heegaard Floer theorists. Since the crossing number is <= (arc index -1)^2/2, this should give better estimates than Lackenby's theorem for some cases of connect sums of knots with small crossing numbers.

    Comment by ianagol — April 20, 2012 @ 11:59 am | Reply

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