# Celebratio Mathematica

## Martin Scharlemann

### A conversation with Martin Scharlemann

#### by Evelyn Lamb

Marty Schar­le­mann was born in 1948. He re­ceived his un­der­gradu­ate de­gree from Prin­ceton Uni­versity in 1969 and his PhD from the Uni­versity of Cali­for­nia, Berke­ley in 1974 un­der the su­per­vi­sion of Ro­bi­on Kirby. He spent most of his ca­reer at the Uni­versity of Cali­for­nia, Santa Bar­bara, with vis­it­ing po­s­i­tions at sev­er­al oth­er in­sti­tu­tions, in­clud­ing the In­sti­tute for Ad­vanced Study (IAS), In­sti­tut des Hautes Études Sci­en­ti­fiques (IHES), Uni­versity of Texas at Aus­tin, Math­em­at­ic­al Sci­ences Re­search In­sti­tute (MSRI), and the Re­search In­sti­tute for Math­em­at­ic­al Sci­ences in Kyoto, Ja­pan. His re­search has probed sev­er­al areas of low-di­men­sion­al to­po­logy, in­clud­ing Hee­gaard split­tings, knot the­ory, and three-man­i­folds.

The fol­low­ing is a con­densed and ed­ited ver­sion of the au­thor’s con­ver­sa­tion with Schar­le­mann in June 2020, about his col­lab­or­a­tions with Abi­gail Thompson and Mi­chael Freed­man.

Lamb: How did you start work­ing with Abi­gail Thompson?

Schar­le­mann: Rob Kirby and Ray Lick­or­ish or­gan­ized these sum­mer meet­ings of to­po­lo­gists in Cam­bridge, Eng­land. And so on one of these — and I’ve for­got­ten ex­actly the year1  — Bill Menasco, who was also a Kirby stu­dent, brought along a gradu­ate stu­dent, Abby Thompson.2 I star­ted talk­ing to her there about knot the­ory, which we were both in­ter­ested in. She was work­ing on her dis­ser­ta­tion, mostly with Bill Menasco. (Her thes­is ad­visor, Ju­li­us Shaneson, was in­ter­ested in oth­er areas.) At some point, we both ended up at the MSRI in Berke­ley.3 Bill was trans­fer­ring over to SUNY Buf­falo and worked with Abby for half the year. Then I came and he asked me if I would work with her for the next half be­cause we were in­ter­ested in many of the same things. And so I did, and I think she fin­ished her thes­is ba­sic­ally there. I was kind of her ad hoc ad­visor. She was in­ter­ested in some of the well-known and deep prob­lems in knot the­ory, things like prop­erty P and the be­ha­vi­or of knots with cross­ing changes and band sums. At the end of the year at MSRI, she had the prob­lem of wheth­er to go to Buf­falo and work with Bill, or we were hav­ing a spe­cial year in to­po­logy at UC­SB: the to­po­lo­gists were al­lowed to hire all the lec­tures for the de­part­ment that year, with the hope that we would get a good group to­geth­er. And so she, in fact, came to Santa Bar­bara. (I’m sure that the dif­fer­ence between the weath­er in Santa Bar­bara and Buf­falo had noth­ing to do with her de­cision!)

Be­cause we were in­ter­ested in the same things, and she had lots of good ideas, we star­ted writ­ing pa­pers to­geth­er. Even­tu­ally, she ended up at UC Dav­is, which is a beau­ti­ful train ride from Santa Bar­bara.

Lamb: Your first pa­per with her is called “Un­knot­ting num­bers, genus, and com­pan­ion tori” [1]. What is that about?

Schar­le­mann: I had be­come very in­ter­ested in something called su­tured man­i­fold the­ory that was de­veloped by Dave Gabai [e5]. I wanted to do it in a slightly dif­fer­ent way, a more com­bin­at­or­i­al way. Dave used fo­li­ations, which didn’t strike me as be­ing at the heart of things, and it wasn’t an area I un­der­stood well. And so I ul­ti­mately was able to do that. Abby had used some the­or­ems in work she did be­fore that [e6], tak­ing some of the out­put of su­tured man­i­fold the­ory to ad­dress prob­lems in knot the­ory. One of the ways that this enters in is in think­ing about cross­ing changes in knots or about band moves. Cross­ing changes are in­tu­it­ive. You just change your cross­ing. Band moves are a little bit more ex­cit­ing. You take a rib­bon from one place in the knot or link and con­nect it to an­oth­er. And then you cut out the middle of the rib­bon.

I think it was in the earli­er pa­per that she real­ized that the the­or­ems that Gabai was cre­at­ing were very rel­ev­ant to ques­tions about band sums and cross­ing changes. And so that first pa­per was an ex­plor­a­tion of that top­ic, how to use su­tured man­i­fold the­or­ems to say things about cross­ings and band sums.

Lamb: I saw that solv­ing the graph planar­ity prob­lem [2] seems to be one of the most im­port­ant prob­lems from your col­lab­or­a­tion. Can you tell me about that?

Schar­le­mann: That came late in our col­lab­or­a­tion, but it was one of the most pleas­ing [pro­jects] be­cause you can ex­plain to any­one what the prob­lem is. You’re giv­en a graph in three-space: ver­tices and arcs con­nect­ing them. And you ask the ques­tion “Can you move it, in three-space, so that it lies on a plane?” There are two as­pects of that. One is the graph prob­lem, which was solved by Kur­atowski [e1]: a graph can­not be put in a plane, even ab­stractly, if and only if it con­tains one of ex­actly two con­fig­ur­a­tions. But bey­ond that, then you ask, “Well, once we know that the graph it­self can be em­bed­ded, can it be moved to the plane in three-space?” And that’s what we were ad­dress­ing.

The simplest ex­ample of that might be just the graph con­sist­ing of a single ver­tex and a single edge, a loop. Of course, that is a planar graph: you just put it in a plane. But it could start out in three-space em­bed­ded as a knot. So then it be­comes the prob­lem: how do you de­cide wheth­er, or can you de­cide wheth­er, a knot in three-space is un­knot­ted, that it can be put in the plane? Well, that was solved by Papakyriako­poulos in the 1950s [e2]. And so our the­or­em could be viewed as an ex­ten­sion of that work, as well. It was a lot of fun. We fairly soon found an idea that seemed very prom­ising via an in­duct­ive step, but it only worked on fairly large graphs. So there was a gen­er­al idea, but then much of the work came in hand­ling the smal­ler graphs as we got closer and closer to the smal­lest in­ter­est­ing one, which would be the Papakyriako­poulos res­ult. Even­tu­ally we got it, and we were very pleased with it.

One of the stor­ies as­so­ci­ated with that is there was ac­tu­ally a bet between two knot ex­perts, Camer­on Gor­don and Jon Si­mon, who pro­posed the prob­lem. It was his con­jec­ture that we proved. They bet \\$20 on wheth­er it would ever be solved. Camer­on be­lieved it would nev­er be solved, and Jon bet against that be­cause he was try­ing to en­cour­age people to do it. I think that Jon Si­mon even­tu­ally col­lec­ted his debt, but you’d have to ask him that.

Lamb: So I guess that wasn’t a bet that ac­tu­ally re­mu­ner­ated the people who solved it.

Schar­le­mann: Yeah, where’s our cut?

Lamb: This might be show­ing my naïveté, but look­ing through your pub­lic­a­tions, I see a lot that I would nat­ur­ally think of as be­ing to­po­logy, but I would have thought of graph planar­ity as be­ing more in the graph the­ory/com­bin­at­or­ics world. So how is this con­nec­ted to what we might think of as clas­sic low-di­men­sion­al to­po­logy?

Schar­le­mann: There is a way of look­ing at the struc­ture of three-man­i­folds called Hee­gaard split­tings, in which you start with what’s called a handle­body. A handle­body looks roughly like a thickened-up bou­quet of circles. And it’s an ele­ment­ary the­or­em, prob­ably go­ing back to the 19th cen­tury, that any closed, com­pact three-man­i­fold can be writ­ten as the uni­on of two such things along their bound­ary.

One way of think­ing about a handle­body is by think­ing of a neigh­bor­hood of a graph. Take any fi­nite graph, put it in three-space, and thick­en that up. If you sup­pose the graph lies in a plane and you thick­en it up, and then you ask, “What does the rest of three-space look like?” it doesn’t take long be­fore you dis­cov­er, well, that’s also the neigh­bor­hood of a graph. And so you’ve de­scribed in that way an ac­tu­al Hee­gaard split­ting of the three-sphere. One part is the handle­body ly­ing near the plane, the oth­er is its com­ple­ment. A kind of fun­da­ment­al the­or­em of Hee­gaard split­tings, due to Wald­hausen [e3], is that for the three-sphere, that’s it: any Hee­gaard split­ting of the three-sphere looks like that. So one up­shot of that the­or­em, if you try and think about it in terms of graphs in three-space, is that if you have a graph sit­ting in three-space, and you know that its com­ple­ment also looks like a thickened-up graph, then you have a Hee­gaard split­ting of the three-space. Wald­hausen tells you that you can take this graph sit­ting in three-space, and if its com­ple­ment is right, then you can slide it around, edges over edges and all that sort of thing, un­til fi­nally it be­comes a nice planar graph sit­ting in the plane. What Jon Si­mon asked was, “What if you don’t al­low your­self to slide edges over edges?” If you just leave it as a graph, it’s ob­vi­ously a more dif­fi­cult ques­tion, and so then the con­jec­ture was if you know that every sub­graph can be put in the plane, and you know that the com­ple­ment is right per Wald­hausen’s the­or­em, and every sub­graph of the ori­gin­al graph has the prop­erty that it can be put in­to a plane — then you can put the graph in the plane. So there are all these con­nec­tions to Hee­gaard the­ory, which Abby and I were quite in­ter­ested in at the time. We worked quite a bit on those kinds of is­sues — Hee­gaard the­ory and knot the­ory. Both of those come to­geth­er in that kind of ques­tion.

Lamb: In your col­lab­or­a­tion with Thompson, did you feel like there was an arc, one big thing you were try­ing to push to, or did you feel like you were al­ways find­ing new prob­lems to work on and just kept me­an­der­ing?

Schar­le­mann: I would say it’s more of the lat­ter. We had tech­niques and were think­ing about prob­lems where they might be ap­plic­able. For ex­ample, this graph planar­ity thing. If my memory is cor­rect, at first we star­ted think­ing about ways we might ap­ply su­tured man­i­fold the­ory to that set­ting. In the end, it wasn’t ne­ces­sary. That the­or­em could have been proven without su­tured man­i­fold the­ory, and that’s the way we ended up do­ing it. So it could have been proven dec­ades ago. I don’t think there was any­thing re­mark­ably mod­ern about the tools we ended up us­ing. It was just put­ting them to­geth­er right.

Lamb: How did you work to­geth­er in this col­lab­or­a­tion — what kind of give and take did you have, how did you bounce ideas off of each oth­er?

Schar­le­mann: I think of her as the ar­chi­tect, and I was the plumb­er. She had these ideas; some of them struck me as com­pletely off the wall, and some of them seemed like they could prob­ably work. (There seemed to be no con­nec­tion between my ini­tial im­pres­sion and how things turned out.) But I have to con­fess I get really ex­cited about the tech­nic­al de­tails of ar­gu­ments and try­ing to fig­ure out ex­actly how to nail things down, and she’s more of the idea per­son. So we kind of worked in that way: she would pitch ideas, and I would try and fig­ure out wheth­er there was an ap­proach from, say, su­tured man­i­fold the­ory. Our roles switched back and forth, but I would say pre­dom­in­antly it was like that.

We had a lot of luck. We got a lot of stuff done, and I was de­lighted at how eas­ily it worked. It seemed like most things we star­ted think­ing about went some­where, and some­where in­ter­est­ing.

Lamb: That is lucky!

Schar­le­mann: As I’m sure you know, 99 per­cent of the things you try in math­em­at­ics don’t work, and some­how you don’t get cred­it for all those great ideas that don’t work!

Lamb: Right. It seems very un­fair. Glan­cing through your pub­lic­a­tion list, there’s a mix of single-au­thor pa­pers and things that you’ve writ­ten with col­lab­or­at­ors. How do you feel like the roles of col­lab­or­a­tion and work­ing by your­self have worked in your ca­reer?

Schar­le­mann: I am prob­ably more of a sol­it­ary work­er than is the norm now, though I think when I star­ted my ca­reer it was more of a norm to work on your own. I like to think in­tens­ively in ways that I of­ten have a very hard time ex­plain­ing to people. In par­tic­u­lar, I can’t draw, and this is a real draw­back if you’re a to­po­lo­gist. I can see pic­tures per­fectly in my head, but I can’t draw, and that makes com­mu­nic­a­tion dif­fi­cult. But one of the nice things about col­lab­or­at­ing with Abby was that she could draw. That’s kind of a stu­pid com­ment, but she also had a way of rein­ing me in when I star­ted bor­ing deep­er than was ne­ces­sary for a giv­en prob­lem, kind of bring­ing me back to real­ity and say­ing, “Why don’t we try this in­stead?” It of­ten seemed that I was en­joy­ing think­ing hard about prob­lems in ways that turned out not to be very pro­duct­ive. She was very help­ful in get­ting me back on track. And she had lots of ideas that would nev­er have oc­curred to me, and then I would get ex­cited about ex­actly how to make those ideas con­crete. A lot of that was sol­it­ary work, but we would re­turn and start talk­ing about things, and it of­ten worked out. I def­in­itely get a lot of en­joy­ment in just think­ing about things hard, walk­ing around and think­ing about them. So I prob­ably am not as col­lab­or­at­ive as many.

Lamb: And how did you start work­ing with Mi­chael Freed­man?

Schar­le­mann: I ran in­to him in his first job, which was at Berke­ley. He ar­rived my fi­nal year in gradu­ate school. We were both in­ter­ested in four-man­i­fold stuff. I fol­lowed with great in­terest what he was do­ing, but we didn’t col­lab­or­ate at that time. The pa­pers I’ve writ­ten with him have been from the past five years. But we talked a lot in the in the 1980s. Over the years, I would see him at con­fer­ences, and we’d some­times have con­ver­sa­tions about four-man­i­folds. Freed­man is in­ter­ested in all kinds of stuff, and he re­mem­bers it very well, too. So my over­lap with him be­came nar­row­er and nar­row­er in terms of his broad­en­ing in­terests. I don’t know if we even ever came close to work­ing on a pa­per to­geth­er [earli­er in our ca­reers]. We would bounce ideas off each oth­er, but it nev­er went any­where, though I learned a lot and I hope he learned some.

As to our re­cent col­lab­or­a­tion, he be­came very in­ter­ested in phys­ics and in quantum the­ory, quantum to­po­logy, et cet­era, which I nev­er really warmed to. At some glor­i­ous mo­ment, he ended up in Santa Bar­bara, not as a math­em­atician in the math de­part­ment, but con­nec­ted to the Mi­crosoft quantum com­put­ing lab. I kind of smile at that be­cause it took way, way too long for us to be able to get things set up so that Mike could be as­so­ci­ated with the de­part­ment. But now he’s nom­in­ally a mem­ber of the UC­SB Math­em­at­ics De­part­ment. So we were geo­graph­ic­ally in the same place, and we talked every once in a while about stuff. You know, I have to say, sit­ting in my of­fice work­ing on these prob­lems, I’d feel a little bit like a mole un­der­ground, and then Mike would walk in, and he’d have all these ideas, and it was like the sun sud­denly shin­ing in­to my tun­nel. And it was it was kind of fun.

But there was a ques­tion called the Pow­ell con­jec­ture. Maybe five or six years ago, Mike got very in­ter­ested in this con­jec­ture be­cause he saw it as con­nec­ted to four-man­i­fold the­ory. The ques­tion that was in­volved there was a kind of ana­log to ques­tions about braid the­ory in three-man­i­fold the­ory. He thought it could be quite im­port­ant in four-man­i­fold the­ory, and so we star­ted talk­ing about that. He sent me over a sketch of a way he thought that you might be able to prove it. I was in the pro­cess of send­ing back an email ex­plain­ing to him why I didn’t think that ap­proach could pos­sibly work when I star­ted think­ing, “Wait a minute! Maybe it could.” And so that’s what launched it. Over the years I think we’re get­ting fairly close to something that looks like proof. It’s still un­clear wheth­er it’s go­ing to all work out, but along the way, we’ve proved some nice stuff that gave rise to the pa­pers you can see. Oth­er stuff is in the 99 per­cent that didn’t work, and so there’s kind of no re­cord of that. But it’s an on­go­ing pro­ject.

There’s an in­ter­est­ing back­ground story to the Pow­ell con­jec­ture. Jerome Pow­ell — not the chair of the Fed­er­al Re­serve, but someone with the same name — was a stu­dent of Joan Birman in the early 1980s and proved a nice the­or­em for his thes­is [e4]. I men­tioned Wald­hausen’s the­or­em about how Hee­gaard split­tings of $$S^3$$ look, but you can ask a sort of second-or­der mov­ing pic­ture ques­tion. Giv­en that you have a Hee­gaard split­ting of $$S^3$$, as Wald­hausen de­scribed, sup­pose you just sort of throw it up in space, and then it comes back down and lands in the same place, but maybe it’s moved around or changed dur­ing that pro­cess. And the ques­tion is, are there simple moves that gen­er­ate all pos­sib­il­it­ies of how this handle­body could move in space to re­turn ex­actly to it­self? Pow­ell proved that five par­tic­u­lar moves suf­fice. Many years later, I’d been think­ing about it, and one of the tech­niques that had worked very well in a lot of set­tings, and in par­tic­u­lar with my col­lab­or­a­tion with Abby Thompson, was the no­tion of thin po­s­i­tion. She proved a really im­port­ant the­or­em in three-man­i­folds, the re­cog­ni­tion prob­lem for the three-sphere, us­ing thin po­s­i­tion [e7]. And I thought, “Well, it’d be nice to come up with a mod­ern proof of Pow­ell’s the­or­em that would use thin po­s­i­tion.” It would cer­tainly be easi­er. But I couldn’t fig­ure out how to do it. I would get stuck at a cer­tain point, and I fi­nally de­cided to “cheat” and read Pow­ell to see how he handled this situ­ation. (I hadn’t really read his pa­per care­fully; I just knew the res­ult.) And I dis­covered there was ac­tu­ally a gap, that he had missed something. There seemed to be no way to pro­ceed, and no one knew how to fix it. I spent a lot of time try­ing to fix it us­ing thin po­s­i­tion, but I still couldn’t do it.

Years later, Mike got in­ter­ested in the prob­lem for the reas­ons I’ve de­scribed, and re­membered that I had been in­ter­ested in it for a while, so that’s why he con­tac­ted me. He asked what I thought of his idea. And it’s a ter­rif­ic idea, but it just isn’t quite enough. And we haven’t quite been able to work it out.

Lamb: Are all your pa­pers with Freed­man re­lated to Pow­ell’s con­jec­ture?

Schar­le­mann: They’re all re­lated to the train of thought ini­ti­ated there. One pa­per is a dis­cus­sion of Mike’s big idea of how Pow­ell’s con­jec­ture might be solv­able [3]. It was known to be true for genus two Hee­gaard split­tings. (That goes back to the 1930s.) And one of our con­crete res­ults there was that the Pow­ell con­jec­ture is also true in genus three. It’s not a big thing, but it was at least a con­crete res­ult. We haven’t sub­mit­ted that pa­per for pub­lic­a­tion any­where yet. We view it as more of a dis­cus­sion doc­u­ment. The hope is that we’ll fig­ure out some way to have a happy con­clu­sion to it, but at the mo­ment, it’s not a sat­is­fact­ory pa­per.

At some point we worked on the Dehn’s lemma pa­per [4]. It was Mike’s idea that this might be true, and he really pushed it through. When I first met him back in the 1970s, he was an ex­pert on Dehn’s lemma and just knew it in and out. He real­ized there was a some­what stronger state­ment you could make, and that just floored me. It’s such an old and hon­or­able the­or­em. That there’s a way of mak­ing it even stronger after all this time really sur­prised me.

The pa­per that comes next on my vita is called “A strong Haken’s The­or­em” [5], which solved a prob­lem that goes back to the 1960s. The ar­gu­ment, to­geth­er with ideas that came out of Freed­man’s ori­gin­al ideas for the Pow­ell con­jec­ture, were what led then to the second col­lab­or­at­ive pa­per with Mike Freed­man, “Unique­ness in Haken’s The­or­em” [6]. It’s a weird co­in­cid­ence that the up­grade of Dehn’s lemma was ex­actly the tool that was needed to make the Strong Haken The­or­em work.

Lamb: Do you feel like the style of your col­lab­or­a­tion with Freed­man is sim­il­ar to your col­lab­or­a­tion with Thompson?

Schar­le­mann: There are sim­il­ar­it­ies, but some­how [work­ing with Freed­man] was al­ways a bit more form­al than work­ing with Abby. Part of that prob­ably has to do with the fact that I was her ad­visor, so I kind of felt like the seni­or fig­ure, where­as it’s clear when I work with Mike that he knows sev­er­al or­ders of mag­nitude more than I do. But I would say in both cases, the col­lab­or­a­tion is pretty com­fort­able. And I have to say I am — maybe I shouldn’t be, but I am — con­cerned about wast­ing his time be­cause he’s do­ing so much in so many dif­fer­ent fields. I don’t want to call him up and say, “I just want to talk about this.” I feel like I have to have something to say.

Lamb: How do you feel look­ing back on what you’ve man­aged to do in this field? Are there still big prob­lems that mo­tiv­ate you?

Schar­le­mann: Oh, I really like big prob­lems. But I go to­wards them with the at­ti­tude that I don’t ne­ces­sar­ily ex­pect to solve them. I would like to un­der­stand them, and that of­ten gives rise to oth­er the­or­ems that I end up find­ing really in­ter­est­ing and can solve. I’ll men­tion the Schoen­flies con­jec­ture as an ex­ample. It is just a beau­ti­ful prob­lem, and think­ing about it, try­ing to fig­ure out how you might try vari­ous ap­proaches, has been really pro­duct­ive for my think­ing. I also like tools that I un­der­stand that are visu­al. I get in­ter­ested in how I can use them to try and view dif­fer­ent prob­lems. Su­tured man­i­fold the­ory is an ex­ample. When I learned about that, it seemed like it was kind of on the bor­der­line of something I might be able to un­der­stand if I worked really hard. And that worked out. I did fi­nally un­der­stand it enough to use it in ori­gin­al ways. Thin po­s­i­tion is an­oth­er ex­ample. A lot of the stuff that Abby and I did was a com­bin­a­tion of those two tools. I star­ted out on this Pow­ell con­jec­ture with ex­actly that in mind. I didn’t think the ques­tion it­self was par­tic­u­larly in­ter­est­ing, but I en­joyed try­ing to think about how this new tool of thin po­s­i­tion might be ap­plic­able to Pow­ell’s the­or­em. Well, it went in totally dif­fer­ent dir­ec­tion when it turns out that the the­or­em hadn’t been proven, and I’ve spent a lot of time work­ing on that. It’s not a Schoen­flies con­jec­ture level prob­lem, but it’s a very in­ter­est­ing prob­lem. And it has led to nice pa­pers: a strong Haken the­or­em pa­per with Mike — well, both of the pa­pers are on an at­tempt to solve the Pow­ell con­jec­ture, and the Dehn’s lemma came out of that. So even though we haven’t solved the prob­lem yet, I think we’ve done a lot of nice math­em­at­ics.

No one has ever asked me that ques­tion be­fore and I’ve just made that up in the last five minutes, but I think it’s pretty ac­cur­ate. I get ex­cited by big prob­lems that I can un­der­stand, and I get ex­cited about new tools that I can use.

Lamb: I’ve heard oth­er math­em­aticians say they got a lot of mileage out of know­ing how to use one par­tic­u­lar tool or ar­gu­ment. It sounds like that’s sort of been your ex­per­i­ence, to some ex­tent: “I know how to do this, so let’s see if it will work.”

Schar­le­mann: I think part of the thing that’s in­ter­est­ing in math­em­at­ics is how dif­fer­ent people can be, and still be suc­cess­ful. I think there are quite a few math­em­aticians for whom what I’ve de­scribed wouldn’t be a nat­ur­al way of think­ing about things. That’s one of the ex­cit­ing things about work­ing in it for so long, is see­ing such a vari­ety of per­son­al­ity types and ap­proaches to math­em­at­ics, and see­ing that they can all be suc­cess­ful in dif­fer­ent ways.

Evelyn Lamb is a freel­ance math and sci­ence writer based in Salt Lake City, Utah. She holds a Ph.D. in math­em­at­ics (Rice Uni­versity, 2012) and has writ­ten for such na­tion­al pub­lic­a­tions as Sci­entif­ic Amer­ic­an, Quanta Magazine, Slate, Sci­ence News and Nautilus.

### Works

[1] M. Schar­le­mann and A. Thompson: “Un­knot­ting num­ber, genus, and com­pan­ion tori,” Math. Ann. 280 : 2 (1988), pp. 191–​205. MR 929535 Zbl 0616.​57003 article

[2] M. Schar­le­mann and A. Thompson: “De­tect­ing un­knot­ted graphs in 3-space,” J. Diff. Geom. 34 : 2 (1991), pp. 539–​560. MR 1131443 Zbl 0751.​05033 article

[3] M. Freed­man and M. Schar­le­mann: Pow­ell moves and the Goer­itz group. Pre­print, 2018. ArXiv 1804.​05909 techreport

[4] M. Freed­man and M. Schar­le­mann: “Dehn’s lemma for im­mersed loops,” Math. Res. Lett. 25 : 6 (2018), pp. 1827–​1836. MR 3934846 article

[5] M. Schar­le­mann: A strong Haken’s The­or­em. Pre­print, 2020. ArXiv 2003.​08523 techreport

[6] M. Freed­man and M. Schar­le­mann: Unique­ness in Haken’s The­or­em. Pre­print, 2020. ArXiv 2004.​07385 techreport