Celebratio Mathematica

My col­lab­or­a­tion with Bob Wil­li­ams began in Janu­ary 1982 and con­tin­ued for about 3–4 years. I refer to him as Bob, be­cause our re­la­tion­ship was in­form­al from day 1. It was both in­ter­est­ing and highly re­ward­ing for both of us, and while we did not work to­geth­er after that, we re­mained good friends, with a friend­ship that had its roots in the pleas­ure that we both felt when we made a new dis­cov­ery in math­em­at­ics.

I had gone to the 1982 winter meet­ing of the AMS for a com­mit­tee meet­ing, but there was free time too, so I wandered over to listen to one of the in­vited talks. Ar­riv­ing early, I sat down next to Bob Wil­li­ams. We in­tro­duced ourselves and began to chat. He told me that he worked in dy­nam­ics, more par­tic­u­larly on chaot­ic flows in dif­fer­en­ti­able dy­nam­ic­al sys­tems, an area about which I knew very little. I told him that I was in­ter­ested in braids, knots and links in three-space. That promp­ted him to dig in­to his pock­et and re­move a piece of pa­per with a pic­ture of what I learned later was a Lorenz knot. It was a simple closed curve in three-space, giv­en by a pro­jec­tion onto a plane, with the six double points. It was easy to see that there were two loops that could be pulled tight to re­move two of the double points (al­beit at the ex­pense of los­ing the in­ter­est­ing pat­tern that was evid­ent in the ori­gin­al pic­ture), but at first glance an eraser was needed to see wheth­er all the double points could be re­moved. He asked “Is it knot­ted?” I answered, al­most im­me­di­ately, “Yes.” He asked me how I knew? I ex­plained, and our col­lab­or­a­tion was on its way!

How did I know? The rep­res­ent­a­tion of knots and links by closed braids had played a cent­ral role in my own re­search, and I could see that the nat­ur­al gen­er­al­iz­a­tion of the knot or link in the pic­ture in Bob’s pock­et was in the form of a (slightly con­cealed) closed braid with all cross­ings hav­ing the same sign. By a stroke of luck, I happened to know that the cross­ing num­ber minus the num­ber of braid strands (3 in his ex­ample) plus 1, was a knot type in­vari­ant (es­sen­tially the genus). In the ex­ample he showed me the genus was 1, not 0 as it would have been if it was the un­knot, and that was how I knew it was not the un­knot.

His ex­ample in­ter­ested me im­me­di­ately, be­cause I had been search­ing for in­ter­est­ing fam­il­ies of knots and links that were giv­en by pro­jec­tions, but not ne­ces­sar­ily by min­im­um cross­ing pro­jec­tions, and it was im­me­di­ately clear that Bob was show­ing me such a fam­ily. A word is in or­der about the clas­sic­al knot tables, re­pro­duced at the end of Dale Rolf­sen’s ex­cel­lent text­book [e2]. The tables list knots by choos­ing a min­im­um cross­ing num­ber planar pro­jec­tion, be­gin­ning with cross­ing num­ber \( 3,4,\dots \) and go­ing up to 8 cross­ings. At the time Bob and I met I knew that three of our col­leagues [e4] were us­ing mod­ern com­puters in an at­tempt to ex­tend the tables in [e2] to \( 9,10,\dots \) cross­ings, only to dis­cov­er that by the time they reached cross­ing num­ber 17 there were 1,701,936 dis­tinct knot types, each defined by a pic­ture! Very little struc­ture was dis­covered. I had be­came con­vinced, at the time when I met Bob, that min­im­um cross­ing num­ber was prob­ably not a good meas­ure of the com­plex­ity of knot type, and Lorenz knots were a beau­ti­ful ex­ample to show that my in­stincts were on the money.

I learned, from Bob, that the nat­ur­al gen­er­al­iz­a­tion of the knot in the pic­ture that he had shown me is the fam­ily of closed or­bits in the solu­tions to a very simple set of three ODEs, with space vari­ables \( x,y,z \) and time vari­able \( t \). The sys­tem is non­lin­ear, but it’s very close to a sys­tem of lin­ear equa­tions in \( x,y,z,t \). Bob ex­plained to me that the solu­tions are a fam­ily of simple closed curves in \( R^3 \) that may be pro­jec­ted, sim­ul­tan­eously and dis­jointly, onto a “branched 2-man­i­fold”, now known as a tem­plate (we called it a knot-hold­er), and even more that the tem­plate de­term­ined an in­fin­ite fam­ily of knot and link dia­grams that we called Lorenz links.

A short time later Bob told me about a second fam­ily of knots that he had be­gun to study, that were also de­term­ined by a tem­plate in \( R^3 \). The fig­ure 8 knot \( F_8 \) em­beds in \( R^3 \), and since \( R^3 \) em­beds in \( S^3 \), the com­ple­ment of \( F_8 \) can be re­garded as a sub­set of \( R^3 \subset S^3 \). He knew that \( F_8 = \partial W^2 \), where \( W^2 \) is a 2-man­i­fold with bound­ary that is em­bed­ded in \( R^3 \), and that \( S^3\setminus F_8 \) has the struc­ture of a sur­face bundle with \( W^2 \) as fiber. It fol­lows that if one can find (as Bob had star­ted to do) a sur­face that it bounds in \( R^3 \), then one should be able to fol­low points on that sur­face as one pushes the fiber off it­self and around in a circle and then back onto it­self, and trace out their or­bits, which would be knots and links in \( R^3 \). When I vis­ited him in his of­fice one day I was as­ton­ished to see that en­tire pro­ject un­der way, a massive con­struc­tion made up out of strips of pa­per and string, held to­geth­er with pa­per clips and glue, and sus­pen­ded from his of­fice ceil­ing! So it turned out that my new friend had been think­ing about knots and links very ser­i­ously for quite some time be­fore we met! See [1], for ex­ample, where he worked out in de­tail the sym­bol­ic dy­nam­ics that al­lowed one to de­term­ine the closed or­bits in the Lorenz at­tract­or, point­ing out that most of the or­bits ap­peared to be knot­ted.

A short time after that ini­tial meet­ing, Bob told me about the tem­plates and as­so­ci­ated branched sur­faces and the tools of sym­bol­ic dy­nam­ics that were at the heart of his work. In turn, I told him about the tools that I knew from knot and link the­ory. Work­ing to­geth­er, we proved (among oth­er things) some in­ter­est­ing prop­er­ties of Lorenz knots and links:

1. There are in­fin­itely many in­equi­val­ent Lorenz knots. These in­clude Lorenz knots of ar­bit­rar­ily high genus, al­though for fixed genus \( g \) only fi­nitely many dis­tinct knot types oc­cur.
2. Every Lorenz knot and link is fibered.
3. Every al­geb­ra­ic knot is a Lorenz knot, and some al­geb­ra­ic links are Lorenz links. In par­tic­u­lar, all tor­us knots oc­cur but some tor­us links do not.
4. There are Lorenz knots that are not it­er­ated tor­us knots; there are it­er­ated tor­us knots that are Lorenz but not al­geb­ra­ic.
5. Every Lorenz link is a closed pos­it­ive braid, however there are closed pos­it­ive braids that are not Lorenz.
6. Every non­trivi­al Lorenz link of two com­pon­ents is un­split­table, also the al­geb­ra­ic and geo­met­ric link­ing num­bers are pos­it­ive and equal.
7. Non­trivi­al Lorenz knots and links are no­namphichir­al.
8. Non­trivi­al Lorenz links have pos­it­ive sig­na­ture.

Most im­port­ant, to me, was the fact that we had dis­covered a fam­ily of knots and links that had im­me­di­ate struc­ture, and they were defined by a fam­ily of link dia­grams that did not have min­im­um cross­ing num­ber. As for Bob, he was the first math­em­atician in his area to sys­tem­at­ic­ally study the knot and link types of the closed or­bits in a dif­fer­en­ti­able dy­nam­ic­al sys­tem, and to show that the knot and link types of the closed or­bits in a dy­nam­ic­al sys­tem had in­de­pend­ent in­terest as a new fam­ily of knots and links.

Fol­low­ing our suc­cess with Lorenz links, we began to study the peri­od­ic or­bits in the com­ple­ment of \( F_8 \), first com­plet­ing Bob’s pro­ject of find­ing its tem­plate, and in­vest­ig­at­ing the fam­ily \( \mathcal F_8 \subset (R^3\setminus F_8) \) of fig­ure 8 knots and links (see [3]). To our as­ton­ish­ment we were un­able to find any res­ults like the ones we quoted above for Lorenz knots and links! We didn’t know what to make of it, un­til Rob Ghrist [e3] poin­ted the way a few years later: Ghrist proved that there are two types of tem­plates as­so­ci­ated to flows in \( R^3 \): In the gen­er­ic case (e.g., the flow as­so­ci­ated to \( \mathcal F_8 \)) the closed or­bits are ALL knots and links. But in very spe­cial cases (e.g., the Lorenz flow) the knots and links are a unique fam­ily, and they char­ac­ter­ize the flow, which is a typ­ic­al ex­ample of a flow that had been dubbed “chaot­ic”, but in fact has a great deal of struc­ture. Moreover the knots and links char­ac­ter­ize the flow. Since Lorenz links had their ori­gins in the ODEs that were used in 1962 to pre­dict the weath­er [e1], that was very in­ter­est­ing.