Narrow Results

For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. An example is also presented of two Legendrian knots that, while topologically isotopic, are distinguished as Legendrian knots by cube number.

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on ${ℝ}^3.$ In particular, we show that the total curvature of a Legendrian knot $L$ in ${ℝ}^3$ is bounded below by $\pi$ times, the total number of cusps in the front projection of $L$. We also show that a Legendrian $(p,q)$-torus knot has the total curvature bounded below by $2\pi q$ while that of the Legendrian knots $E(l,k)$ is bounded below by $2\pi (k+l+1)$. Furthermore, we find an explicit relation between the Thurston–Bennequin number of a Legendrian knot $L$ and the geometric self-linking number, the curvature and the torsion of the knot $L$.

We prove that every Legendrian knot in the tight contact structure of the $3$-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight $3$-sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given.

This note concerns Legendrian cobordisms in one-jet spaces of functions, in the sense of Arnol’d [Lagrange and Legendre cobordisms. I, Funkt. Anal. Prilozhen. 14(3) (1980) 1–13, 96.] — consisting of big Legendrian submanifolds between two smaller ones. We are interested in such cobordisms which fit with generating functions, and wonder which structures and obstructions come with this notion. As a central result, we show that the classes of Legendrian concordances with respect to the generating function equipment can be given a group structure. To this construction, we add one of a homotopy with respect to generating functions.

We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots ${{𝒦}}_m$, for $m\ge 3$, satisfy the Uniform Thickness Property.

In this paper, mosaic tiles have been used to build knots and define invariants since they were introduced by Lomonaco and Kauffman to describe quantum knots. In this paper, we propose a modified set of tiles to describe the front projections of Legendrian knots. We explore the effect of stabilization on the mosaic number of a Legendrian knot by defining and classifying “two-cell” stabilizations. Inspired by a construction of Ludwig et al. in 2013, we provide an infinite family of Legendrian unknots whose mosaic numbers are realized only in non-reduced projections. Finally, we provide a census of known mosaic numbers for Legendrian unknots and trefoils.

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya’s combinatorial description of the Heegaard Floer contact invariant.

In a recent work of Dynnikov and Prasolov [I. Dynnikov and M. Prasolov, An algorithm for comparing Legendrian knots, preprint (2023), arXiv:2309.05087] a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method, we confirm conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997] about Legendrian knots in topological types $7_4$, $9_{48}$ and $10_{136}$. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.

An invariant polynomial s(t) is defined for framed knots in the solid torus. The coefficients are Vassiliev invariants of order one. An invariant polynomial A(t) of Legendrian curves is introduced and it is shown how to calculate it from their fronts. The coefficient of A(t) of the order n term is the restriction to the discriminant of the selftangencies with partial index n of the Arnold invariant J⁺ of wave fronts. The polynomial A(t) of a Legendrian curve is recovered from the polynomial s(t) of the Legendrian knot, provided with its natural contact framing.

A subset of legendrian 2-string tangles are defined to be minimal if the strands realize the minimum absolute value of the Bennequin invariant. The restrictiveness of this condition is examined by studying which topological rational tangles have minimal representatives. It is shown that a rational of finite parity has a minimal representative if and only if its standard continued fraction expansion contains only non-negative entries, is of odd length, and has every horizontal entry even. This is proved by applying recent results of Fuchs and Tabachnikov or Chmutov and Goryunov that give an upperbound for the Bennequin invariant of links in terms of the minimal exponent of the framing variable of the Kauffman polynomial and Yokota's precise formula for this topological invariant. A second geometric stratum of rational tangles with infinite parity is defined and it is then shown a positive rational has a minimal represenative if and only if its continued fraction expansion is a particular extension of a minimal of finite parity and a negative rational of infinite parity has a minimal representative if and only if it has even length with every vertical entry even.

For a Legendrian knot L ⊂ ℝ³, 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 F, 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.