# Celebratio Mathematica

## Dmitry Fuchs

### Dmitry Fuchs and Legendrian knot theory

#### 1. Legendrian and transverse knots

The ba­sic ob­jects of study are as fol­lows: A con­tact struc­ture on a 3-man­i­fold is a max­im­ally non­in­teg­rable 2-plane field. A Le­gendri­an (resp. trans­verse) knot in a con­tact 3-man­i­fold, $$(M, \xi)$$, is a smoothly em­bed­ded closed curve in $$M$$ that is every­where tan­gent (resp. every­where trans­verse) to $$\xi$$. Le­gendri­an and trans­verse knots play a key role in 3- and 4-di­men­sion­al con­tact and sym­plect­ic to­po­logy sim­il­ar to that of knots and links in low-di­men­sion­al to­po­logy, e.g., al­low­ing for con­struc­tion of man­i­folds via sur­gery/handle at­tach­ment or branched cov­er­ing. Moreover, as in­dic­ated by Ben­nequin’s work (see Sec­tion 2.2 be­low) and the tight/over­twisted di­cho­tomy sub­sequently dis­covered by Eli­ash­berg, the be­ha­vi­or of Le­gendri­an knots can re­flect prop­er­ties of the con­tact man­i­folds that they pop­u­late.

An im­port­ant case is when $$M=\mathbb{R}^3$$ with its stand­ard con­tact struc­ture, $\xi_{\mathrm{std}} = \operatorname{ker}( dz-y\,dx)$ (as pic­tured in Fig­ure 1), where the study of Le­gendri­an knots can be viewed as an in­ter­est­ing vari­ant on the clas­sic­al the­ory of knot­ted circles in 3-di­men­sion­al space. In this set­ting, a smooth knot $$L \subset \mathbb{R}^3$$ para­met­rized as $$t \mapsto (x(t),y(t),z(t))$$ is Le­gendri­an if and only if it sat­is­fies the dif­fer­en­tial equa­tion $$z^{\prime}(t) = y(t) x^{\prime}(t)$$. As this amounts to the iden­tity $$y = dz/dx$$, Le­gendri­an knots in stand­ard con­tact $$\mathbb{R}^3$$ are con­veni­ently viewed via their front pro­jec­tions to the $$xz$$-plane since the miss­ing $$y$$-co­ordin­ate can be re­covered as the slope in this pro­jec­tion. Front pro­jec­tions of Le­gendri­an knots are closed curves without tan­gen­tial double points or ver­tic­al tan­gen­cies and hav­ing cusp sin­gu­lar­it­ies; see Fig­ure 1. In the re­mainder of the art­icle, un­less oth­er­wise spe­cified, all Le­gendri­an knots are in $$(\mathbb{R}^3, \xi_{\mathrm{std}})$$.

Two Le­gendri­an knots are Le­gendri­an iso­top­ic if they are smoothly iso­top­ic through oth­er Le­gendri­an knots. A sim­il­ar no­tion of trans­verse iso­topy ex­ists for trans­verse knots and either no­tion is equi­val­ent to the knots be­ing re­lated by an am­bi­ent con­tact iso­topy. Note that any Le­gendri­an knot has a well-defined un­der­ly­ing to­po­lo­gic­al type as an or­din­ary (smooth) knot in $$\mathbb{R}^3$$, so that one can speak for in­stance of Le­gendri­an un­knots or Le­gendri­an tre­foils. A fun­da­ment­al prob­lem of Le­gendri­an knot the­ory is:

The Le­gendri­an iso­topy prob­lem: Clas­si­fy Le­gendri­an knots of a giv­en to­po­lo­gic­al knot type up to Le­gendri­an iso­topy.

The clas­si­fic­a­tion for Le­gendri­an un­knots was ac­com­plished by Eli­ash­berg and Fraser in [e11], [e31]. Around 2000, Le­gendri­an tor­us knots and fig­ure-eight knots were clas­si­fied by Et­nyre and Honda [e15], and sub­sequently com­plete clas­si­fic­a­tions have been ob­tained for some ad­di­tion­al fam­il­ies of to­po­lo­gic­al knot types; see, e.g., [e39], [e37]. In gen­er­al, the Le­gendri­an iso­topy prob­lem re­mains dif­fi­cult. For in­stance, a glance at the Le­gendri­an Knot At­las [e38] re­veals many to­po­lo­gic­al knot types with 9 or few­er cross­ings con­tain­ing Le­gendri­an knots that are con­jec­tured to be dis­tinct but have not been suc­cess­fully dis­tin­guished with any known in­vari­ants.

#### 2. The work of Fuchs and Tabachnikov

In the past dec­ades there has been something of an ex­plo­sion of work re­lated to Le­gendri­an knot the­ory. In­deed, in June 2020 Math­S­ciNet re­turns 133 matches for art­icles con­tain­ing “Le­gendri­an” and “knot” or “link” in the title. The sixth of these to ap­pear chro­no­lo­gic­ally, and one of the most highly cited, is the art­icle “In­vari­ants of Le­gendri­an and trans­verse knots in the stand­ard con­tact space”, by D. Fuchs and S. Tabach­nikov.

##### 2.1. Stable classification

A fun­da­ment­al res­ult from [1] ad­dresses a stable ver­sion of the Le­gendri­an iso­topy prob­lem. For a Le­gendri­an knot, $$L \subset \mathbb{R}^3$$, pos­it­ive and neg­at­ive sta­bil­iz­a­tions of $$L$$, de­noted by $$S_{+}(L)$$ and $$S_-(L)$$, arise from adding zig-zags to the front pro­jec­tion of $$L$$, as shown in Fig­ure 2, where the sign is de­term­ined by the wheth­er the ori­ent­a­tion of the knot passes the new cusps in the down­ward or up­ward dir­ec­tion.

The­or­em 2.1: ([1])  If $$L_1$$ and $$L_2$$ are Le­gendri­an knots with the same to­po­lo­gic­al knot type, then after ap­ply­ing some num­ber of pos­it­ive and neg­at­ive sta­bil­iz­a­tions to each of $$L_1$$ and $$L_2$$ they will be­come Le­gendri­an iso­top­ic.

Thus, the stable Le­gendri­an iso­topy prob­lem re­duces to the iso­topy prob­lem for to­po­lo­gic­al knots!1 The idea of the proof is to con­vert the se­quence of to­po­lo­gic­al knot dia­grams that ap­pear dur­ing a gen­er­ic to­po­lo­gic­al iso­topy in­to front pro­jec­tions of Le­gendri­an knots by adding cusps at ver­tic­al tan­gen­cies and near cross­ings where the over strand has lar­ger slope. The suc­cess­ive to­po­lo­gic­al dia­grams that ap­pear are re­lated by Re­idemeister moves and oth­er modi­fic­a­tions in­volving ver­tic­al tan­gen­cies, and it is shown that for any such bi­furc­a­tion the cor­res­pond­ing front pro­jec­tions will be re­lated by Le­gendri­an iso­topy after adding enough zig-zags.

In The­or­em 2.1, it is im­port­ant that sta­bil­iz­a­tions of both signs are al­lowed. In­deed, in the art­icle by Ep­stein, Fuchs and Mey­er [2] a modi­fic­a­tion of the ar­gu­ment from [1] is used to show that two Le­gendri­an knots be­come equi­val­ent after pos­it­ive (resp. neg­at­ive) sta­bil­iz­a­tions if and only if their pos­it­ive (resp. neg­at­ive) trans­verse push-offs, ob­tained by shift­ing a small amount in the pos­it­ive (resp. neg­at­ive) nor­mal dir­ec­tion with­in the con­tact planes, are iso­top­ic as trans­verse knots. As a res­ult, the iso­topy prob­lem for trans­verse knots is re­duced to the $$S_+$$-stable (or $$S_-$$-stable) ver­sion of the Le­gendri­an iso­topy prob­lem.

##### 2.2. Bennequin-type inequalities and knot polynomials

In di­men­sion 3, there is an op­por­tun­ity for in­ter­ac­tion between Le­gendri­an knot the­ory and to­po­lo­gic­al knot the­ory. In this dir­ec­tion, Fuchs and Tabach­nikov ob­served in [1] an in­ter­est­ing re­la­tion between the clas­sic­al in­vari­ants of a Le­gendri­an knot and the fam­ous HOM­FLY-PT and Kauff­man knot poly­no­mi­als2 dis­covered in the 1980s. Be­fore stat­ing their res­ult, let us re­view the con­text.

There are two clas­sic­al in­teger-val­ued in­vari­ants of a Le­gendri­an knot $$L$$: the Thur­ston–Ben­nequin num­ber, $$\operatorname{\mathit{tb}}(L)$$, meas­ures the link­ing num­ber of $$L$$ with its pos­it­ive trans­verse push-off, and the ro­ta­tion num­ber, $$r(L)$$, meas­ures ro­ta­tion of the tan­gent vec­tor to $$L$$ with­in the con­tact planes us­ing a trivi­al­iz­a­tion of $$\xi$$. With the clas­sic­al in­vari­ants in mind, a start­ing point for the Le­gendri­an iso­topy prob­lem is to try to an­swer the fol­low­ing prob­lem for­mu­lated by Eli­ash­berg in [e6]:

The Le­gendri­an geo­graphy prob­lem: What pairs of in­tegers can be real­ized by the val­ues $$(\operatorname{\mathit{tb}}(L), r(L))$$ for a Le­gendri­an knot $$L$$ in a giv­en to­po­lo­gic­al knot type $$\mathcal{K}$$?

By ap­ply­ing sta­bil­iz­a­tions, it is easy to make $$\operatorname{\mathit{tb}}(L)$$ be­come neg­at­ive with ar­bit­rar­ily large mag­nitude without chan­ging the to­po­lo­gic­al knot type. However, for the stand­ard (tight!) con­tact struc­ture on $$\mathbb{R}^3$$ the Thur­ston–Ben­nequin num­ber is bounded above with­in any fixed to­po­lo­gic­al knot type. The first such up­per bound ap­pears in the sem­in­al work of Ben­nequin [e1] which es­tab­lished the ex­ist­ence of a con­tact struc­ture on $$\mathbb{R}^3$$ not dif­feo­morph­ic to the stand­ard one. Ben­nequin proved an in­equal­ity for trans­verse knots that, by con­sid­er­ing trans­verse push-offs, is equi­val­ent to the state­ment that any Le­gendri­an knot $$L$$ in $$(\mathbb{R}^3, \xi_{\mathrm{std}})$$ sat­is­fies $\operatorname{\mathit{tb}}(L) + |r(L)| \leq 2g(L)-1,$ where $$g(L)$$ is the min­im­um genus of any Seifert sur­face for $$L$$. In par­tic­u­lar, any Le­gendri­an un­knot must have $$\operatorname{\mathit{tb}} \leq -1$$. By con­trast, us­ing the con­tact struc­ture $$\xi_{\mathrm{ot}}$$ giv­en in cyl­indric­al co­ordin­ates as the ker­nel of the 1-form $$\cos r\, dz+ r \sin r\,d\theta$$ the circle in the $$xy$$-plane centered at the ori­gin with ra­di­us $$2\pi$$ is a Le­gendri­an un­knot with $$\operatorname{\mathit{tb}}=0$$. Thus, there is no dif­feo­morph­ism of $$\mathbb{R}^3$$ tak­ing $$\xi_{\mathrm{std}}$$ to $$\xi_{\mathrm{ot}}$$. From Ben­nequin’s in­equal­ity, we see that there is a max­im­um value of $$\operatorname{\mathit{tb}}$$ among Le­gendri­an knots in any to­po­lo­gic­al knot type $$\mathcal{K}$$ that we will de­note by $$\overline{\operatorname{\mathit{tb}}}(\mathcal{K})$$. Note that ex­cept for un­knots the right-hand side of the in­equal­ity is pos­it­ive, and this raises the ques­tion: are there any to­po­lo­gic­al knot types with $$\overline{\operatorname{\mathit{tb}}}(\mathcal{K})$$ less than $$-1$$?

Fuchs and Tabach­nikov found that in­equal­it­ies of a sim­il­ar fla­vor to Ben­nequin’s arise from the HOM­FLY-PT and Kauff­man knot poly­no­mi­als, $$P_K$$ and $$F_K$$. These in­vari­ants as­sign to any to­po­lo­gic­al knot $$K$$ two vari­able Laurent poly­no­mi­als, $$P_K, F_K \in \mathbb{Z}[a^{\pm1}, z^{\pm1}]$$, that are char­ac­ter­ized by simple skein re­la­tions. For in­stance, the HOM­FLY-PT poly­no­mi­al is uniquely de­term­ined by iso­topy in­vari­ance, its value on the un­knot which we take here to be 1, and the iden­tity shown in Fig­ure 3, which holds when a knot dia­gram is mod­i­fied loc­ally as pic­tured.

The­or­em 2.2: ([1])  For any Le­gendri­an link $$L \subset (\mathbb{R}^3, \xi_{\mathrm{std}})$$, we have the in­equal­it­ies $\operatorname{\mathit{tb}}(K) +|r(K)| \leq -\operatorname{deg}_a P_K(a,z) -1$ and $$\operatorname{\mathit{tb}}(K) \leq - \operatorname{deg}_a F_{K}(a,z) -1,$$ where $$\operatorname{deg}_a P_K$$ and $$\operatorname{deg}_a F_K$$ de­note the max­im­um de­gree in the vari­able $$a$$ of the HOM­FLY-PT and Kauff­man poly­no­mi­als.

In [1], the HOM­FLY-PT in­equal­ity is ob­served as an im­me­di­ate con­sequence of the com­bin­a­tion of the works of Ben­nequin and Mor­ton, Franks and Wil­li­ams [e2], [e3], and the Kauff­man es­tim­ate is then de­duced us­ing a nice trick in­volving Rudolph’s re­la­tion [e5] between the Kauff­man poly­no­mi­al of a knot and the HOM­FLY-PT poly­no­mi­al of its double with op­pos­ite ori­ent­a­tions.3 As such, the au­thors of [1] elec­ted to loc­ate these in­equal­it­ies in the back­ground sec­tion, pos­sibly lead­ing to some later con­fu­sion in the lit­er­at­ure about the ori­gin of these in­equal­it­ies.

For the left-handed tre­foil, $$T$$, a simple com­pu­ta­tion of the Kauff­man poly­no­mi­al re­solves the above ques­tion, as $F_T(a,z) = a^2(2+z^2) -a^3z -a^4(1+z^2) + a^5z,$ so that The­or­em 2.2 gives a sharp es­tim­ate show­ing that $$\overline{\operatorname{\mathit{tb}}}(T) = -6$$ is in­deed less than $$-1$$.4 Moreover, either of the es­tim­ates from The­or­em 2.2 gives an ele­ment­ary proof (per­haps the simplest) that $$(\mathbb{R}^3, \xi_{\mathrm{std}})$$ is tight and thus is dis­tinct from $$\xi_{\mathrm{ot}}$$. In later years, sev­er­al ad­di­tion­al Ben­nequin-type in­equal­it­ies have been dis­covered with quant­it­ies de­rived from vari­ous to­po­lo­gic­al knot in­vari­ants ap­pear­ing on the right-hand side. See [e29] for the state of the art circa 2008, as well as [e10], [e36], [e40] for ex­ten­sions of The­or­em 2.2 to some oth­er am­bi­ent con­tact man­i­folds.

##### 2.3. Legendrian simplicity and finite-type invariants

To com­ple­ment the Le­gendri­an geo­graphy prob­lem, it is nat­ur­al to ask to what ex­tent the clas­sic­al in­vari­ants de­term­ine Le­gendri­an (or trans­verse) knot types.5 In this dir­ec­tion, call a to­po­lo­gic­al knot type Le­gendri­an (or trans­vers­ally) simple if any two Le­gendri­an (or trans­verse) knots with­in the knot type that have equal clas­sic­al in­vari­ants are Le­gendri­an (or trans­verse) iso­top­ic. At the time [1] was writ­ten, it was not known wheth­er any knot types in $$(\mathbb{R}^3, \xi_{\mathrm{std}})$$ were nonsimple in either the Le­gendri­an or trans­verse sense.6 One way to pro­duce ex­amples of Le­gendri­an knots with the same clas­sic­al in­vari­ants and un­der­ly­ing knot type is by the Le­gendri­an mir­ror op­er­a­tion that takes $$L$$ to its im­age $$L^\vee$$ un­der the dif­feo­morph­ism of $$\mathbb{R}^3$$ that maps $$(x,y,z) \mapsto (x,-y,-z)$$. See Fig­ure 4. In gen­er­al, $$\operatorname{\mathit{tb}}(L^\vee) = \operatorname{\mathit{tb}}(L)$$ and $$r(L^\vee) = -r(L)$$, so when $$r(L) =0$$ the clas­sic­al in­vari­ants are un­changed lead­ing to the fol­low­ing prob­lem for­mu­lated by Fuchs and Tabach­nikov in [1].

Le­gendri­an mir­ror prob­lem: Do there ex­ist Le­gendri­an knots with $$r(L)=0$$ such that $$L$$ and $$L^\vee$$ are not Le­gendri­an iso­top­ic?

At the time, the cent­ral dif­fi­culty in find­ing ex­amples of non-Le­gendri­an simple knot types was that no Le­gendri­an in­vari­ants bey­ond $$\operatorname{\mathit{tb}}$$ and $$r$$ were known to ex­ist. One of the main res­ults of [1], which uses a nat­ur­al ex­ten­sion of the class of fi­nite-type knot in­vari­ants (see, e.g., [e8]) to the set­ting of Le­gendri­an and trans­verse knots, elu­cid­ates the dif­fi­culty of find­ing such in­vari­ants and could even be viewed as evid­ence that such in­vari­ants might not ex­ist.7

The­or­em 2.3: Two Le­gendri­an (or trans­verse) knots with the same clas­sic­al in­vari­ants can­not be dis­tin­guished by any fi­nite-type Le­gendri­an (or trans­verse) in­vari­ant.

This the­or­em was later ex­ten­ded to Le­gendri­an knots in a much wider class of con­tact man­i­folds, in­clud­ing all tight con­tact struc­tures, in the work of V. Tch­ernov [e18]. In­ter­est­ingly, Tch­ernov also finds ex­amples of knots in over­twisted con­tact struc­tures on $$S^1\times S^2$$, where a state­ment ana­log­ous to that of The­or­em 2.3 no longer holds!

#### 3. New invariants and normal rulings

In the late 1990s a ma­jor de­vel­op­ment oc­curred in Le­gendri­an knot the­ory with the dis­cov­ery8 of the Chekan­ov–Eli­ash­berg al­gebra, a new in­vari­ant cap­able of dis­tin­guish­ing between Le­gendri­an knots with the same clas­sic­al in­vari­ants. This in­vari­ant is a dif­fer­en­tial graded al­gebra (ab­bre­vi­ated DGA) arising from $$J$$-holo­morph­ic curve the­ory that can be for­mu­lated in an en­tirely com­bin­at­or­i­al man­ner. In its ini­tial ver­sion,9 the DGA as­signed to a Le­gendri­an knot $$L$$ con­sists of a free as­so­ci­at­ive (but non­com­mut­at­ive) $$\mathbb{Z}/2\mathbb{Z}$$-al­gebra, $$\mathcal{A}(L)$$, gen­er­ated by the double points of $$L$$ un­der the Lag­rangi­an pro­jec­tion, $$\pi_{xy}:\mathbb{R}^3 \rightarrow \mathbb{R}^2$$, $$(x,y,z) \mapsto (x,y)$$, to­geth­er with a dif­fer­en­tial $$\partial: \mathcal{A}(L) \rightarrow \mathcal{A}(L)$$ defined by a count of im­mersed poly­gons in the plane with bound­ary on $$\pi_{xy}(L)$$. A $$\mathbb{Z}/2r(L)\mathbb{Z}$$-grad­ing, which proves cru­cial in many ap­plic­a­tions, arises from the ro­ta­tion num­bers of cer­tain cap­ping paths for cross­ings of $$\pi_{xy}(L)$$. In his fam­ous art­icle [e16], Chekan­ov ap­plied the DGA to dis­tin­guish a pair of $$m(5_2)$$ knots with the same clas­sic­al in­vari­ants, es­tab­lish­ing the first ex­ample of a Le­gendri­an nonsimple knot type. As an­oth­er early ap­plic­a­tion, in [e19] Lenny Ng made use of the non­com­mut­ativ­ity of the Chekan­ov–Eli­ash­berg DGA to re­solve the Le­gendri­an mir­ror prob­lem by ex­hib­it­ing a Le­gendri­an $$6_2$$ knot, pic­tured in Fig­ure 5, with zero ro­ta­tion num­ber that is not equi­val­ent to its Le­gendri­an mir­ror.10

##### 3.1. Augmentations and normal rulings

Fuchs was an early pro­ponent of the Chekan­ov–Eli­ash­berg DGA, as he ex­plored and ap­plied this in­vari­ant in a se­quence of art­icles sev­er­al of which were coau­thored with grad stu­dents at UC Dav­is. The Chekan­ov–Eli­ash­berg DGA is some­what un­wieldy to work with in its en­tirety as the graded pieces of $$\mathcal{A}(L)$$ and also of its ho­mo­logy tend to be in­fin­ite-di­men­sion­al. One way to ob­tain eas­ily com­put­able in­vari­ants, used ini­tially by Chekan­ov, is to em­ploy an aug­ment­a­tion of $$\mathcal{A}(L)$$ which is an al­gebra map $$\epsilon: \mathcal{A}(L) \rightarrow \mathbb{Z}/2\mathbb{Z}$$ that sat­is­fies $$\epsilon \circ \partial = 0$$, $$\epsilon(1) =1$$, and pre­serves grad­ing. When $$\mathcal{A}(L)$$ has an aug­ment­a­tion, the DGA can be lin­ear­ized to pro­duce a fi­nite-di­men­sion­al chain com­plex whose ho­mo­logy can be used for dis­tin­guish­ing Le­gendri­an knots. For in­stance, in [2] Ep­stein, Fuchs, and Mey­er es­tab­lish the ex­ist­ence of aug­ment­a­tions for a fam­ily of Le­gendri­an twist knots, re­ferred to there as Eli­ash­berg knots, and are then able to ef­fort­lessly dis­tin­guish these knots from one an­oth­er us­ing only the de­gree dis­tri­bu­tion of cross­ings.11 An im­port­ant ques­tion then be­comes:

Ques­tion 3.1: For a Le­gendri­an knot $$L \subset \mathbb{R}^3$$, when does $$(\mathcal{A}(L), \partial)$$ have an aug­ment­a­tion?

In in­vest­ig­at­ing this ques­tion, Fuchs dis­covered in [4] a beau­ti­ful com­bin­at­or­i­al struc­ture arising in the front pro­jec­tions of Le­gendri­an knots called a nor­mal rul­ing. For a Le­gendri­an link $$L$$ in $$\mathbb{R}^3$$, a nor­mal rul­ing is a de­com­pos­i­tion of the front pro­jec­tion of $$L$$ in­to a col­lec­tion of simple closed curves, each of which has corners at a left and right cusp, and at some sub­set of the cross­ings called “switches”; see Fig­ure 5. The de­com­pos­i­tion is sub­ject to cer­tain re­stric­tions, in­clud­ing the nor­mal­ity con­di­tion il­lus­trated in Fig­ure 5. Giv­en a di­visor, $$\rho \,|\, 2 r(L)$$, a nor­mal rul­ing is said to be $$\rho$$-graded if all cross­ings that are switches have their de­grees con­gru­ent to 0 mod $$\rho$$. One can also con­sider $$\rho$$-graded aug­ment­a­tions by weak­en­ing the grad­ing re­quire­ment so that $$\epsilon:\mathcal{A}(L) \rightarrow \mathbb{Z}/2$$ only needs to pre­serve grad­ing mod $$\rho$$. Of­ten 1-graded nor­mal rul­ings or aug­ment­a­tions are re­ferred to as un­graded since when $$\rho=1$$ the grad­ing con­di­tion be­comes vacu­ous.

The­or­em 3.2: ([4], [5], [e21]) . The Chekan­ov–Eli­ash­berg DGA of a Le­gendri­an knot $$L$$ has a $$\rho$$-graded aug­ment­a­tion if and only if the front pro­jec­tion of $$L$$ has a $$\rho$$-graded nor­mal rul­ing.

The for­ward dir­ec­tion was es­tab­lished in [4], while the re­verse im­plic­a­tion was proven in­de­pend­ently by Fuchs and Ishkhan­ov [5] and Sabloff [e21]. For the proof, in [4] Fuchs in­tro­duced an el­eg­ant meth­od of adding “splashes” via a Le­gendri­an iso­topy in or­der to greatly sim­pli­fy the dif­fer­en­tial at the ex­pense of adding many ad­di­tion­al gen­er­at­ors to $$\mathcal{A}(L)$$. Fuchs’ splashes ap­pear as little ripples in the front pro­jec­tion that be­come steep waves when viewed in the Lag­rangi­an pro­jec­tion. As a res­ult the holo­morph­ic disks that con­trib­ute to the dif­fer­en­tial of $$(\mathcal{A}(L), \partial)$$ be­come trapped in thin ver­tic­al strips between suc­cess­ive pairs of splashes. This res­ults in ex­pli­cit mat­rix for­mu­las for dif­fer­en­tials de­pend­ing only on the loc­al ap­pear­ance of $$L$$ between two splashes, which may be taken to con­sist of a single cross­ing or cusp.12

In­ter­est­ingly, the no­tion of nor­mal rul­ing was dis­covered in­de­pend­ently by Chekan­ov and Pushkar in a some­what dif­fer­ent con­text.13 In [e20] they in­tro­duced nor­mal rul­ings, un­der the name of pos­it­ive prop­er de­com­pos­i­tions, in the con­text of gen­er­at­ing fam­il­ies for Le­gendri­an knots, and ap­plied prop­er­ties of the be­ha­vi­or of nor­mal rul­ings un­der gen­er­ic Le­gendri­an iso­top­ies in their solu­tion of the Arnold 4-con­jec­tures.14 A gen­er­at­ing fam­ily (of func­tions) for a Le­gendri­an knot $$L$$ in $$\mathbb{R}^3$$ is a one-para­met­er fam­ily of func­tions $$\{f_x\}_{x \in \mathbb{R}}$$ whose crit­ic­al val­ues trace out the front pro­jec­tion of $$L$$. Gen­er­at­ing fam­il­ies are a stand­ard tool in sym­plect­ic to­po­logy, and they were ap­plied in the work of Traynor [e14] to dis­tin­guish cer­tain two-com­pon­ent Le­gendri­an links with the same clas­sic­al in­vari­ants. In in­flu­en­tial but un­pub­lished work, Pushkar had also sug­ges­ted a meth­od for de­fin­ing a Le­gendri­an ho­mo­logy us­ing gen­er­at­ing fam­il­ies as well as a com­bin­at­or­i­al ap­proach to the whole the­ory via what M. B. Henry would call Morse com­plex se­quences in his thes­is [e32]. As (i) Chekan­ov and Pushkar had shown that (lin­ear at in­fin­ity) gen­er­at­ing fam­il­ies for $$L$$ ex­ist if and only if $$L$$ has a nor­mal rul­ing, and (ii) Pushkar’s gen­er­at­ing fam­ily ho­mo­logy could be com­puted us­ing a com­plex gen­er­ated by the cross­ings of $$\pi_{xy}(L)$$, it seemed likely that there would be a con­nec­tion between gen­er­at­ing fam­il­ies and aug­ment­a­tions.

In [6], Fuchs and I es­tab­lished such a con­nec­tion by con­struct­ing an aug­ment­a­tion from a gen­er­at­ing fam­ily and provid­ing an iso­morph­ism between the cor­res­pond­ing lin­ear­ized ho­mo­logy and gen­er­at­ing fam­ily ho­mo­logy groups. Un­der this iso­morph­ism, Sabloff’s du­al­ity res­ult [e28] for the lin­ear­ized ho­mo­logy groups of the Chekan­ov–Eli­ash­berg al­gebra ap­pears on the gen­er­at­ing fam­ily side as the Al­ex­an­der du­al­ity. The idea for con­struct­ing an aug­ment­a­tion, $$\epsilon$$, from a gen­er­at­ing fam­ily, $$\{f_x\}$$, is to again work with a splashed ver­sion of $$L$$. Then, the gen­er­at­ors of $$\mathcal{A}(L)$$ can be col­lec­ted to­geth­er in a se­quence of matrices $$X_1, \ldots, X_r$$ and $$Y_1, \ldots, Y_r$$ as­so­ci­ated to the dif­fer­ent splashes loc­ated at a se­quence of $$x$$-val­ues $$x_1 < x_2 < \cdots < x_r$$. One then uses the gen­er­at­ing fam­ily to define an aug­ment­a­tion, $$\epsilon$$, by tak­ing $$\epsilon(Y_i)$$ to be the mat­rix of the dif­fer­en­tial in the Morse com­plex of $$f_{x_i}$$ and $$\epsilon(X_i)$$ to be the mat­rix of a con­tinu­ation map between the Morse com­plexes of $$f_{x_i}$$ and $$f_{x_{i+1}}$$. The lin­ear­ized ho­mo­logy com­plex as­so­ci­ated to $$\epsilon$$ then cor­res­ponds to the gen­er­at­ing fam­ily ho­mo­logy via a cel­lu­lar­iz­a­tion by fiber­wise des­cend­ing/as­cend­ing man­i­folds that was also known to Pushkar as a means for com­put­ing the gen­er­at­ing fam­ily ho­mo­logy.

##### 3.2. Fuchs’ “irresponsible conjecture”

In the art­icle [4] where he in­tro­duced nor­mal rul­ings, Fuchs made an el­eg­ant con­jec­ture con­nect­ing the new Le­gendri­an in­vari­ants with to­po­lo­gic­al knot the­ory.

Con­jec­ture 3.3: An un­graded aug­ment­a­tion for the Chekan­ov–Eli­ash­berg al­gebra, $$(\mathcal{A}(L),\partial)$$, should ex­ist if and only if the Kauff­man poly­no­mi­al es­tim­ate for $$\operatorname{\mathit{tb}}(L)$$ from The­or­em 2.2 is sharp.

The con­jec­ture was based on a study of mir­ror tor­us knots in [3] and evid­ence from knots with small cross­ing num­ber. Des­pite be­ing dubbed “ir­re­spons­ible” in [4], the con­jec­ture turned out to be com­pletely ac­cur­ate and was proven in [e25] as a con­sequence of a more pre­cise re­la­tion­ship between nor­mal rul­ings and the Kauff­man poly­no­mi­al. In [e20], Chekan­ov and Pushkar had ob­tained Le­gendri­an in­vari­ants by mak­ing a re­fined count of nor­mal rul­ings: they showed that for any $$n \in \mathbb{Z}$$ and $$\rho \,|\, 2 r(L)$$, the num­ber, $$f^\rho_n$$, of $$\rho$$-graded nor­mal rul­ings, $$\sigma$$, of $$L$$ with $j(\sigma) := \#\,\mbox{switches} - \#\,\mbox{right cusps}$ equal to $$n$$ is a Le­gendri­an iso­topy in­vari­ant of $$L$$. With $$\rho$$ fixed, it is con­veni­ent to col­lect these num­bers as the coef­fi­cients of a $$\rho$$-graded rul­ing poly­no­mi­al defined by sum­ming over all $$\rho$$-graded nor­mal rul­ings for $$L$$ as $R_L^\rho(z):= \sum_{\sigma} z^{j(\sigma)} = \sum_{n\in \mathbb{Z}} f^\rho_{n}z^n.$ The 0-graded rul­ing poly­no­mi­al can dis­tin­guish knots with the same clas­sic­al in­vari­ants, such as Chekan­ov’s pair of $$m(5_2)$$ knots. In con­trast, when $$\rho =1$$ or 2 the rul­ing poly­no­mi­al de­pends only on $$\operatorname{\mathit{tb}}$$ and the un­der­ly­ing to­po­lo­gic­al knot type as shown by the fol­low­ing.

The­or­em 3.4: ([e25])  For any Le­gendri­an link $$L \subset \mathbb{R}^3$$, the 1-graded rul­ing poly­no­mi­al $$R^1_L(z)$$ is the coef­fi­cient of $$a^{-\operatorname{\mathit{tb}}(L)-1}$$ in the Kauff­man poly­no­mi­al, $$F_L(a,z)$$. The 2-graded rul­ing poly­no­mi­al $$R^2_L(z)$$ is the coef­fi­cient of $$a^{-\operatorname{\mathit{tb}}(L)-1}$$ in the HOM­FLY-PT poly­no­mi­al, $$P_L(a,z)$$.

To see that Fuchs’ con­jec­ture fol­lows as a co­rol­lary, ob­serve that the in­equal­ity (1) is sharp if and only if the coef­fi­cient of $$a^{-\operatorname{\mathit{tb}}(L)-1}$$ in $$F_L$$ is nonzero. Since this coef­fi­cient is the un­graded rul­ing poly­no­mi­al, $$R^1_L(z)$$, it is nonzero if and only if $$L$$ has an un­graded nor­mal rul­ing and this is equi­val­ent by The­or­em 3.2 to $$\mathcal{A}(L)$$ hav­ing an un­graded aug­ment­a­tion.

The con­nec­tion between nor­mal rul­ings, aug­ment­a­tions of the Chekan­ov–Eli­ash­berg al­gebra, and the to­po­lo­gic­al knot poly­no­mi­als has been strengthened in sev­er­al sub­sequent works. Build­ing on the many-to-one cor­res­pond­ence between aug­ment­a­tions and nor­mal rul­ings over $$\mathbb{Z}/2\mathbb{Z}$$ from [e26] and Henry’s study of aug­ment­a­tions and Morse com­plex se­quences from [e32], Henry and I showed in [e43] that the rul­ing poly­no­mi­als spe­cial­ized at $$z= q^{1/2}-q^{-1/2}$$ with $$q$$ a prime power cor­res­pond to counts of aug­ment­a­tions of $$\mathcal{A}(L)$$ to fi­nite fields $$\mathbb{F}_q$$ and in this way are de­term­ined by the Chekan­ov–Eli­ash­berg DGA. The works [e41], [e44], [e45] re­late counts of high­er-di­men­sion­al rep­res­ent­a­tions of $$\mathcal{A}(L)$$ to rul­ing poly­no­mi­als of Le­gendri­an satel­lites and (when $$\rho = 1$$ or 2) to the $$n$$-colored Kauff­man and HOM­FLY-PT poly­no­mi­als. Pre­cisely how much of the DGA $$(\mathcal{A}(L), \partial)$$ is de­term­ined by the Thur­ston–Ben­nequin num­ber and to­po­lo­gic­al type of $$L$$ re­mains an in­ter­est­ing ques­tion, and some open con­jec­tures on this top­ic ap­pear in [e19], [e41].

#### 4. Working with Fuchs at UC Davis

Hav­ing Dmitry Fuchs as an ad­visor at UC Dav­is was a priv­ilege and a pleas­ure. I came in­to con­tact with Dmitry, whom I al­ways re­ferred to as Dr. Fuchs while I was a stu­dent, as soon as classes star­ted upon my ar­rival at UC Dav­is in 2003 as he taught the first two thirds of the year-long al­gebra se­quence for new Ph.D. stu­dents. The top­ics covered by Fuchs in the course were lin­ear al­gebra, rings and mod­ules, cat­egory the­ory, and Galois the­ory. Ho­mo­lo­gic­al al­gebra was covered in the fi­nal third of the course se­quence, taught that year by Dmitry’s own ad­visor, Al­bert Schwarz. I par­tic­u­larly en­joyed the ex­er­cises Fuchs had as­sembled for the class, from which stu­dents were al­lowed to choose from an ample se­lec­tion of prob­lems which pulled in many ad­di­tion­al top­ics, es­pe­cially in the cat­egory the­ory part of the class.

As the first year wrapped up, in or­der to re­ceive sum­mer sup­port from UC Dav­is’ VI­GRE grant I needed to work on a re­search pro­ject. But, I had no re­search pro­gram. What to do? With an­oth­er stu­dent, Chris Berg, we de­cided to talk to Fuchs who agreed to su­per­vise a sum­mer pro­ject, and helped us to quickly write a pro­pos­al. Al­though the pro­pos­al had in­volved char­ac­ter­ist­ic classes of fam­il­ies of fo­li­ations, lead­ing me to scramble to as­semble some back­ground in dif­fer­en­ti­able man­i­folds which I did not have at the time, dur­ing the sum­mer Dmitry shared his con­jec­ture about nor­mal rul­ings and the Kauff­man poly­no­mi­al. Fol­low­ing Dmitry’s sug­ges­tions, we es­tab­lished the sharp­ness of the es­tim­ate (1) for sev­er­al fam­il­ies of Le­gendri­an knots con­struc­ted so that their front dia­grams would have ob­vi­ous nor­mal rul­ings.

After the sum­mer, Chris turned his in­terests to­ward com­bin­at­or­i­al rep­res­ent­a­tion the­ory and began work­ing with M. Vazir­ani, go­ing on to do ex­cel­lent work in this area. I con­tin­ued to work with Dr. Fuchs, and as I at­ten­ded his year-long al­geb­ra­ic to­po­logy course,15 we star­ted weekly meet­ings that would con­tin­ue off and on for the rest of my time at UC Dav­is. Dur­ing these meet­ings, which of­ten took place at Dmitry’s home of­fice and could last for 2 to 3 hours, Dmitry was ex­tremely gen­er­ous with his time and know­ledge. As a res­ult, I was ex­posed to a vari­ety of Dmitry’s (many) fa­vor­ite top­ics in­clud­ing Lie al­gebra co­homo­logy and fo­li­ations, evol­utes and in­vol­utes of plane curves, con­tact and sym­plect­ic to­po­logy, rep­res­ent­a­tions of the Vi­ra­s­oro al­gebra, sin­gu­lar vec­tors in Verma mod­ules of af­fine Lie al­geb­ras, etc. Dmitry has a par­tic­u­lar af­fin­ity for (and en­cyc­lo­ped­ic know­ledge of!) clas­sic­al al­geb­ra­ic to­po­logy, and a ques­tion about K-the­ory or spec­tral se­quences could lead to a two-hour im­pro­vised over­view of the top­ic.

With the more spe­cif­ic state­ment in hand, the con­jec­ture was not hard to prove. View­ing the rul­ing in­vari­ant as a knot poly­no­mi­al, I soon real­ized a ver­sion of its skein re­la­tion us­ing a loc­al pic­ture with two cusps. But, it seemed too spe­cif­ic to be ap­plic­able to ar­bit­rary Le­gendri­an knots. Later in my of­fice I saw that a more gen­er­al skein re­la­tion held and checked through the case-by-case in­duct­ive ar­gu­ment that showed that the skein re­la­tion, also sat­is­fied by the top term of the Kauff­man poly­no­mi­al, would uniquely char­ac­ter­ize a Le­gendri­an in­vari­ant. I sent Dmitry a quick e-mail let­ting him know that his con­jec­ture could now be con­sidered a the­or­em. It was ex­cit­ing to know that the res­ult was true!

By gen­er­ously shar­ing his con­jec­ture and oth­er prob­lems with me, Dmitry gave me the chance to enter the math­em­at­ic­al re­search com­munity, but his sup­port did not end there. After fin­ish­ing my Ph.D. at UC Dav­is, Dmitry helped me to ob­tain valu­able post-doc po­s­i­tions at Duke Uni­versity where I worked with Lenny Ng and later at Uni­versity of Arkan­sas where I worked with Yo’av Rieck. Moreover, he helped me to make im­port­ant con­nec­tions with oth­er re­search­ers in­ter­ested in Le­gendri­an knots and re­lated top­ics by, not long after I com­pleted my Ph.D., or­gan­iz­ing with S. Tabach­nikov and L. Traynor a work­shop at AIM. This work­shop also res­ul­ted in an off­shoot SQuaREs group that stim­u­lated my col­lab­or­a­tion with Brad Henry. Dmitry also en­cour­aged me to travel in­ter­na­tion­ally, and in par­tic­u­lar to spend a pro­duct­ive and en­joy­able month at MPIM in Bonn.

Thank you, Dmitry! I am truly grate­ful to you for shar­ing your know­ledge and joy in do­ing math­em­at­ics and for all of your help along the way!

The au­thor is an As­so­ci­ate Pro­fess­or at Ball State Uni­versity in Muncie, In­di­ana. He en­joys spend­ing time with fam­ily and groov­ing to the sounds of clas­sic jazz from the 1950s and 60s.

### Works

[1] D. Fuchs and S. Tabach­nikov: “In­vari­ants of Le­gendri­an and trans­verse knots in the stand­ard con­tact space,” To­po­logy 36 : 5 (1997), pp. 1025–​1053. MR 1445553 Zbl 0904.​57006 article

[2] J. Ep­stein, D. Fuchs, and M. Mey­er: “Chekan­ov–Eli­ash­berg in­vari­ants and trans­verse ap­prox­im­a­tions of Le­gendri­an knots,” Pac. J. Math. 201 : 1 (2001), pp. 89–​106. MR 1867893 Zbl 1049.​57005 article

[3] J. Ep­stein and D. Fuchs: “On the in­vari­ants of Le­gendri­an mir­ror tor­us links,” pp. 103–​115 in Sym­plect­ic and con­tact to­po­logy: In­ter­ac­tions and per­spect­ives (Toronto and Montreal, 26 March–7 April 2001). Edi­ted by Y. Eli­ash­berg, B. Khes­in, and F. Lalonde. Fields In­sti­tute Com­mu­nic­a­tions 35. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2003. MR 1969270 Zbl 1044.​57007 incollection

[4] D. Fuchs: “Chekan­ov–Eli­ash­berg in­vari­ant of Le­gendri­an knots: Ex­ist­ence of aug­ment­a­tions,” J. Geom. Phys. 47 : 1 (July 2003), pp. 43–​65. MR 1985483 Zbl 1028.​57005 article

[5] D. Fuchs and T. Ishkhan­ov: “In­vari­ants of Le­gendri­an knots and de­com­pos­i­tions of front dia­grams,” Mo­sc. Math. J. 4 : 3 (July–September 2004), pp. 707–​717. To Borya Fei­gin, with love. MR 2119145 Zbl 1073.​53106 article

[6] D. Fuchs and D. Ruther­ford: “Gen­er­at­ing fam­il­ies and Le­gendri­an con­tact ho­mo­logy in the stand­ard con­tact space,” J. To­pol. 4 : 1 (2011), pp. 190–​226. MR 2783382 Zbl 1237.​57026 ArXiv 0807.​4277 article