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

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 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 arise from adding
zig-zags to the front projection of __\( S_-(L) \)__,__\( L \)__, 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.

__([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]:

__\( (\operatorname{\mathit{tb}}(L), r(L)) \)__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.

__([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].

__\( r(L)=0 \)__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

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:

__\( L \subset \mathbb{R}^3 \)__, 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.

__([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.

__\( (\mathcal{A}(L),\partial) \)__, 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.

__([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.*