## Three knot invariants

This has been a long Monday, toward the end of which I confused knot width with its bridge number. This post is meant to remind me which is which. Here is a trefoil knot:

The crossing number is the smallest number of self-intersections that the knot can have when thrown on a plane. For the trefoil cr=3.

Here is the same knot on which points of locally maximal height are marked with red, and points of locally minimal height are in blue. There are two extrema of each kind.

The bridge number of a knot is the smallest number of maxima in any of its diagrams. The trefoil has b=2. A knot is in the bridge position if its diagram achieves b, and in addition all the maxima are above all the minima. This allows one to neatly cut the knot by its bridge sphere (in orange), leaving b untangled arcs hanging from either side.

The third invariant also has to do with the extrema of the height function. Now we pick a regular value in each interval between each pair of consecutive critical values, and count the multiplicity of said value. The sum of all these multiplicities is the width of the knot. For the trefoil w=8.

The knot is in a thin position when it attains w, as on the diagram above. Despite their apparent similarity, two different positions of the knot may be required to attain b and w. Also, the invariants behave in different ways under connected sums. Here is a connected sum of two knots, with trefoil on the left:

It’s obvious that the crossing number is subadditive: $\mathrm{cr}\,(K\# K')\le \mathrm{cr}\,(K)+\mathrm{cr}\,(K')$. It remains an open problem whether it’s in fact additive, i.e., whether the equality always holds.

The bridge number is the best behaved of the three: it’s known that $b(K\# K')=b(K)+b(K')-1$ (one-sided inequality is easy: connected sum can kill the absolute maximum on the lower-placed knot). Horst Schubert proved this equality in 1954. Much more recently (in 2004) Jennifer Schultens used modern machinery to give a 6-page proof of Schubert’s theorem.

The width turns out to misbehave. By putting one of the summands way below the other, one sees at once that $w(K\# K')\le w(K)+w(K')-2$. In many examples this is in fact an equality. However, very recently (2011) Blair and Tomova proved that strict inequality may hold; moreover, the Scharlemann-Schultens lower bound $w(K\# K')\ge \min(w(K),w(K'))$ is attained for infinitely many pairs of knots.