Celebratio Mathematica

Dmitry Fuchs

Dmitry Fuchs and Legendrian knot theory

by Dan Rutherford

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 \( (\mathbb{R}^3, \xi_{\mathrm{std}}) \).

Figure 1. Left: the standard contact structure on \( \mathbb{R}^3 \). Right: the front projection of a left-handed Legendrian trefoil.

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
Figure 2.

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 \( -1 \)?

Figure 3.

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 \begin{equation} \operatorname{\mathit{tb}}(K) \leq - \operatorname{deg}_a F_{K}(a,z) -1, \end{equation} 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?
Figure 4. A Legendrian \( 6_2 \) and its Legendrian mirror.

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?
Figure 5. Examples of normal rulings for some Legendrian knots with topological knot type \( m(5_2) \), \( 6_2 \), and \( m(8_{21}) \). The normality condition is the requirement that near switches the vertical intervals connecting the crossing strands to their companion strands (that belong to the same closed curve of the ruling) should either be disjoint or nested.

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

Figure 6. Fuchs’ splashes as they appear in the front and Lagrangian projections.

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.

I was still some­what hooked on the prob­lem about nor­mal rul­ings and the Kauff­man poly­no­mi­al and from time to time would come back to it and ex­pand the class of ex­amples for which I could prove Fuchs’ con­jec­ture. A par­tic­u­lar ex­cit­ing day of math­em­at­ics for me res­ul­ted after read­ing about Chekan­ov and Pushkar’s rul­ing in­vari­ant in the pa­per [e26] of Ng and Sabloff, for­mu­lated there as a multis­et of in­tegers, i.e., a fi­nite se­quence of non­neg­at­ive in­tegers. On the bike ride home I star­ted think­ing about the top de­gree of the Kauff­man poly­no­mi­al, the part that needs to be nonzero in or­der for the es­tim­ate to be sharp. It’s also a se­quence of in­tegers (but non­neg­at­ive?). What if it was the same se­quence? I had a pretty good idea in my head of what the Kauff­man poly­no­mi­al of the tre­foil looked like, and that seemed to check out. Once I got home, I checked the fig­ure-eight knot. As it can be a little tricky to identi­fy all of the nor­mal rul­ings of a front pro­jec­tion by hand, I sat down and wrote a C++ pro­gram to com­pute the rul­ing in­vari­ant and then star­ted com­par­ing the res­ults with the tables of the Kauff­man poly­no­mi­al one knot at a time. Ex­cept for some signs in the coef­fi­cients of the Kauff­man poly­no­mi­al, each and every knot worked out ex­actly!!!16 I was ex­cited to tell Dr. Fuchs the news. At the start of our next meet­ing I let him know that I had made an in­ter­est­ing dis­cov­ery. Dmitry’s re­sponse, with a bit of twinkle in his eye, was “I like in­ter­est­ing dis­cov­er­ies…” He handed me the pen. After un­der­stand­ing my re­fine­ment of his con­jec­ture and check­ing it for a few knots, Dmitry was vis­ibly ex­cited. He im­me­di­ately ad­journed the meet­ing say­ing that he had some oth­er top­ics that he had planned to dis­cuss with me but now it did not seem so im­port­ant. I left his house un­der or­ders to be­gin writ­ing a pa­per with the spe­cial cases that I was able to prove. On the way home I can re­mem­ber paus­ing on a hill in a park and feel­ing sat­is­fied and ex­cited in the per­fect Dav­is sum­mer weath­er.

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.


[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