Celebratio Mathematica

Martin Scharlemann

My favorite theorems

by Martin Scharlemann

Rob Kirby asked me about my fa­vor­ite the­or­ems. I set out to pick a top three but ended up with four, lis­ted in his­tor­ic­al se­quence be­low. The first and the third would be on any list, for the reas­ons I’ve de­scribed; the choice of the oth­er two was hard and might be dif­fer­ent next month. (e.g., my work with Ru­bin­stein would be an­oth­er good choice.)

1. “Smooth spheres in \( \mathbb{R}^4 \) with four crit­ic­al points are stand­ard”, In­ven­tiones Math. 79 (1985) 125–141.

Reas­on: I had been work­ing mostly on 4-man­i­folds when Nich­olas Kuiper asked me if I knew how to show this. I real­ized pretty quickly it was really a 3-di­men­sion­al prob­lem, one that I then learned many had tried to prove. But I had an idea, based largely on my earli­er com­bin­at­or­i­al work on out­er­most forks (in “Tun­nel num­ber one knots sat­is­fy the Poen­aru con­jec­ture” To­po­logy and its Ap­plic­a­tions 18 (1984) 235–258) and it took only a month or so of hard thought to real­ize that it would prob­ably work. So I had thereby solved an old prob­lem in a field (3-man­i­folds) that I knew little about. The meth­ods also im­me­di­ately led to a proof that un­knot­ting num­ber one knots are prime, an­oth­er old prob­lem. Oth­ers then found the core idea in it (even­tu­ally to be called “Schar­le­mann cycles”) use­ful. Time to switch to 3-man­i­folds!

2. “Pro­du­cing re­du­cible 3-man­i­folds by sur­gery on a knot”, To­po­logy 29 (1990) 481–500.

Reas­on: The main take-away of this ar­gu­ment, that satel­lite knots sat­is­fy the cabling con­jec­ture, is less than thrill­ing, but the ar­gu­ment lead­ing to it was prob­ably the deep­est I’ve done. It fea­tures ex­tens­ive use of the su­tured man­i­fold the­ory we had just learned about from Gabai, ap­plied IMHO in a pretty ima­gin­at­ive way. (An earli­er draft claimed to prove the cabling con­jec­ture in gen­er­al, but Dave im­me­di­ately spot­ted an er­ror).

3. (With Abby Thompson) “De­tect­ing un­knot­ted graphs in 3-space”, J. Diff. Geom. 34 (1991) 539–560.

Reas­on: This pa­per solved the graph-planar­ity prob­lem, show­ing that there is an al­gorithm to de­cide wheth­er a fi­nite graph in 3-space can be moved in 3-space in­to the plane. I like it be­cause a) its state­ment is prob­ably the easi­est among my the­or­ems to ex­plain to a non-math­em­atician, and b) it solved a long-stud­ied ques­tion that had even giv­en rise to a bet between Camer­on Gor­don and Jonath­an Si­mon as to wheth­er it would ever be solved.

4. (With Maggy To­mova) “Al­tern­ate Hee­gaard genus bounds dis­tance”, Geo­metry and To­po­logy 10 (2006) 593–617.

There are ac­tu­ally three reas­ons I like this one: a) I thought it was a pretty sur­pris­ing res­ult; b) I heard later that An­drew Cas­son, one of my math idols, had con­jec­tured that something like this might be true and prob­ably had thought about it hard him­self; and c) Wow! I have an­oth­er great stu­dent!