Related papers: On Legendrian Graphs
We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if…
We introduce the concept of link-irregular labelings for graphs, extending the notion of link-irregular graphs through edge labeling with positive integers. A labeling is link-irregular if every vertex has a uniquely labeled subgraph…
A graph $X$ is said to be "unstable" if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if…
We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$,…
For any compact connected submanifold $K$ of $\mathbb{R}^n$, let $\Lambda_K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of $\mathbb{R}^n$. In this paper, we give examples of pairs…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…
Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this bound to…
We compute an explicit $e$-positive formula for the chromatic symmetric function of a lollipop graph, $L_{m,n}$. From here we deduce that there exist countably infinite distinct $e$-positive, and hence Schur-positive, bases of the algebra…
In the unit cotangent bundle of $\mathbb{R}^3$, we consider loops of Legendrian tori arising as families of the unit conormal bundles of smooth knots in $\mathbb{R}^3$. In this paper, using the cord algebra of knots, we give a topological…
We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the…
The Bollob\'as-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored…
It was shown in arXiv:1208.5742 that any smooth knot can be represented by an \"ubercrossing projection, i.e. a knot projection with no crossings aside from a single multi-crossing. We extend this idea to Legendrian knots and investigate…
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. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain…
A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by…
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the…
The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial'…
A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The…
To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is…
Karo\'nski, {\L}uczak and Thomason conjectured in 2004 that for every finite graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$ in such a way that the endvertices of each edge have different sums of incident…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…