Mikhail Lavrov, Dan Rutherford
In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ which is a rational function in variables $a$ and $s$ and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in $S^1\times S^2$ and shows that the invariant is in fact a Laurent polynomial in $a$ and $z= s -s^{-1}$. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston-Bennequin number to Legendrian links in $S^1\times S^2$ with its tight contact structure.
Yu Pan, Dan Rutherford
For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed exact Lagrangian cobordisms by considering their Legendrian lifts as conical Legendrian cobordisms. To a conical Legendrian cobordism $Σ$ from $Λ_-$ to $Λ_+$, we associate an immersed DGA map, which is a diagram $$\alg(Λ_+) \stackrel{f}{\rightarrow} \alg(Σ) \stackrel{i}{\hookleftarrow} \alg(Λ_-), $$ where $f$ is a DGA map and $i$ is an inclusion map. This construction gives a functor between suitably defined categories of Legendrians with immersed Lagrangian cobordisms and DGAs with immersed DGA maps. In an algebraic preliminary, we consider an analog of the mapping cylinder construction in the setting of DG-algebras and establish several of its properties. As an application we give examples of augmentations of Legendrian twist knots that can be induced by an immersed filling with a single double point but cannot be induced by any orientable embedded filling.
Honghao Gao, Dan Rutherford
We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel-Ekholm-Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the first Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist.
Yu Pan, Dan Rutherford
For a Legendrian link $Λ\subset J^1M$ with $M = \mathbb{R}$ or $S^1$, immersed exact Lagrangian fillings $L \subset \mbox{Symp}(J^1M) \cong T^*(\mathbb{R}_{>0} \times M)$ of $Λ$ can be lifted to conical Legendrian fillings $Σ\subset J^1(\mathbb{R}_{>0} \times M)$ of $Λ$. When $Σ$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation $α: \mathcal{A}(Σ) \rightarrow \mathbb{Z}/2$ of the LCH algebra of $Σ$, there is an induced augmentation $ε_{(Σ,α)}: \mathcal{A}(Λ) \rightarrow \mathbb{Z}/2$. With $Σ$ fixed, the set of homotopy classes of all such induced augmentations, $I_Σ\subset \mathit{Aug}(Λ)/{\sim}$, is a Legendrian isotopy invariant of $Σ$. We establish methods to compute $I_Σ$ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary $n \geq 1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $ρ\neq 1$ and $Λ\subset J^1\mathbb{R}$ every $ρ$-graded augmentation of $Λ$ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of $ρ$-graded augmented Legendrian cobordism.
Dan Rutherford
We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.
Michael B. Henry, Dan Rutherford
For a Legendrian knot L in R^3 with a chosen Morse complex sequence (MCS) we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov-Eliashberg DGA after changing coordinates by an augmentation.
Dan Rutherford
We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.
Justin Murray, Dan Rutherford
For any Legendrian knot $K$ in standard contact ${\mathbb R}^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce an ungraded $n$-colored ruling polynomial, $R^1_{n,K}(q)$, as a linear combination of reduced ruling polynomials of positive permutation braids and show that (i) $R^1_{n,K}(q)$ arises as a specialization $F_{n,K}(a,q)\big|_{a^{-1}=0}$ of the $n$-colored Kauffman polynomial and (ii) when $q$ is a power of two $R^1_{n,K}(q)$ agrees with the total ungraded representation number, $\operatorname{Rep}_1\big(K, \mathbb{F}_q^n\big)$, which is a normalized count of $n$-dimensional representations of $(\mathcal{A}(K),\partial)$ over the finite field $\mathbb{F}_q$. This complements results from [Leverson C., Rutherford D., Quantum Topol. 11 (2020), 55-118, arXiv:1802.10531] concerning the colored HOMFLY-PT polynomial, $m$-graded representation numbers, and $m$-graded ruling polynomials with $m \neq 1$.
Dmitry Fuchs, Dan Rutherford
We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\Rightarrow$ normal ruling.
Mikhail Lavrov, Dan Rutherford
For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.
Michael B. Henry, Dan Rutherford
For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the case q =2, we show the augmentation numbers, Aug_m(L,q), are determined by specializing the m-graded ruling polynomial, R^m_L(z), at z = q^{1/2}-q^{-1/2}. As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.
Lenhard Ng, Dan Rutherford
We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological type and Thurston-Bennequin number of the knot.
Yu Pan, Dan Rutherford
We consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $\mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $\mathbb{F}_2$ this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $\dim M = 2$ realizing any prescribed monodromy representation, $Φ:π_1(M,x_0) \rightarrow \mathit{GL}(\mathbf{n}, \mathbb{F})$.
Michael B. Henry, Dan Rutherford
Let L be a Legendrian knot in R^3 with the standard contact structure. In [10], a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of L and have isomorphic linearized contact homology groups. A second corollary states that the count of equivalence classes of Morse complex sequences of a Legendrian knot is a Legendrian isotopy invariant.
Dan Rutherford, Michael G Sullivan
We give a computation of the Legendrian contact homology (LCH) DGA for an arbitrary generic Legendrian surface $L$ in the $1$-jet space of a surface. As input we require a suitable cellular decomposition of the base projection of $L$. A collection of generators is associated to each cell, and the differential is given by explicit matrix formulas. In the present article, we prove that the equivalence class of this cellular DGA does not depend on the choice of decomposition, and in the sequel [35] we use this result to show that the cellular DGA is equivalent to the usual Legendrian contact homology DGA defined via holomorphic curves. Extensions are made to allow Legendrians with non-generic cone-point singularities. We apply our approach to compute the LCH DGA for several examples including an infinite family, and to give general formulas for DGAs of front spinnings allowing for the axis of symmetry to intersect $L$.
Dan Rutherford, Michael Sullivan
This article is a continuation of [15]. For Legendrian surfaces in $1$-jet spaces, we prove that the Cellular DGA defined in [15] is stable tame isomorphic to the Legendrian contact homology DGA.
Dan Rutherford, Michael G. Sullivan
Given an augmentation for a Legendrian surface in a $1$-jet space, $Λ\subset J^1(M)$, we explicitly construct an object, $\mathcal{F} \in Sh_Λ$, of the (derived) category from arXiv:1402.0490 of constructible sheaves on $M\times R$ with singular support determined by $Λ$. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in $1$-jet spaces that, based on arXiv:1608.02984 and arXiv:1608.03011, is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of arXiv:1402.0490 for $1$-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $Sh_Λ$ in the $2$-dimensional case.
Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, Eric Zaslow
We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.
Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek
We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.
Dan Rutherford, Michael G Sullivan
We study augmentations of a Legendrian surface $L$ in the $1$-jet space, $J^1M$, of a surface $M$. We introduce two types of algebraic/combinatorial structures related to the front projection of $L$ that we call chain homotopy diagrams (CHDs) and Morse complex $2$-families (MC2Fs), and show that the existence of either a $ρ$-graded CHD or MC2F is equivalent to the existence of a $ρ$-graded augmentation of the Legendrian contact homology DGA to $\mathbb{Z}/2$. A CHD is an assignment of chain complexes, chain maps, and homotopy operators to the $0$-, $1$-, and $2$-cells of a compatible polygonal decomposition of the base projection of $L$ with restrictions arising from the front projection of $L$. An MC2F consists of a collection of formal handleslide sets and chain complexes, subject to axioms based on the behavior of Morse complexes in $2$-parameter families. We prove that if a Legendrian surface has a tame at infinity generating family, then it has a $0$-graded MC2F and hence a $0$-graded augmentation. In addition, continuation maps and a monodromy representation of $π_1(M)$ are associated to augmentations, and then used to provide more refined obstructions to the existence of generating families that (i) are linear at infinity or (ii) have trival bundle domain. We apply our methods in several examples.