Related papers: Studying uniform thickness I: Legendrian simple it…

We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph…

We present a unified framework to systematically embed complex knotted and linked structures, beyond the torus family, into diverse topological phases, including Hopf insulators, classical spin liquids, topological semimetals, and…

Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory,…

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus…

The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…

Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geq\alpha_0…

Using the Bordered Floer theory of Lipshitz-Ozsv\'ath-Thurston we prove that the $(p,q)$-cables of any non-trivial knots are not Heegaard Floer homologically thin. Using the proof and a theorem of Zemke, we find a larger set of satellite…

In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its…

We show there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot…

We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing…

We show that the genus problem for alternating knots with $n$ crossings has linear time complexity and is in Logspace$(n)$. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We…

We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot and its…

We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display non-local entanglement. To this end, we introduce and study a…

We extend the sutured framework to the case of Legendrians with boundary. Using ideas from Lagrangian Floer theory, we define the cylindrical and the wrapped sutured Legendrian homologies of a pair of sutured Legendrians. They fit together…

There are several knot invariants in the literature that are defined using singular instantons. Such invariants provide strong tools to study the knot group and give topological applications. For instance, it gives powerful tools to study…

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in…

We study the Ozsv\'{a}th-Szab\'{o}-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables $\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $\mathscr{L}_{p,q}$ are constructed…

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots…

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot…

Topological insulators are typically characterized by particularly stable properties, such as global invariants, and can be identified by probing their robust surface states. A recently discovered novel form of band topology, delicate…