arXiv research

Two new minor minimal intrinsically chiral graphs identified.

This paper classifies chiral graphs up to size 12.

We classify which complete multipartite graphs are intrinsically chiral.

This paper identifies all topological symmetry groups for Heawood family graphs.

We introduce invariants of spatial graphs related to the Wu invariant and the Simon invariant, and apply them to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of embedded graphs.

The main result of this paper is that for every closed, connected, orientable, irreducible 3-manifold $M$, there is an integer $n_M$ such that any abstract graph with no automorphism of order 2 which has a 3-connected minor whose genus is more than $n_M$ has no achiral embedding in $M$. By contrast, the paper also pro…

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-…

A new method integrates forms on Riemann surfaces, leading to modular forms.

Survey of intrinsically linked or knotted graphs.

New graph shows edge deletion/contraction doesn't always result in intrinsically linked graphs.

We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families, we verify a conjecture presented in Adams' "The Knot Book": If a vertex is remo…

We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted. We also classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respe…

New research finds six bipartite intrinsically knotted graphs with 23 edges.

We show that deleting an edge of a 3-cycle in an intrinsically knotted graph gives an intrinsically linked graph.

Spatial graphs study tangle replacement with equivalence classes.

Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the thirteen graphs obtained from K7 by rY moves are intrinsically knotted graphs with 21 edges. We prove that these 14 graphs are the only intrinsically knotted graphs with 21 edges.

Chiral string integrands simplify to ambitwistor string integrands in the tensionless limit.

A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are the only minor minimal intrinsically knotted graphs with 21 edges. This set incl…

Paper restricts chirally cosmetic surgeries on knots.

3-manifolds are chiral if not finitely covered by sphere or product.

Intrinsic formulation of noncommutative geometry for quantum gravity.

Graphs and their complements are intrinsically knotted.

We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of the graph contains a non-splittable 2-component link. We also show that there exis…

Study on embeddings and their topological properties in R^d.

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Peterse…

Study shows stable graphs in Heisenberg group are essentially planes.

We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S^3. Also, assuming the Poincare Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S^3.

The paper studies maps in the Heisenberg group and their images, called Rickman rugs.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$…

GCPNet improves molecular graph learning for protein structure and binding.

The paper studies harmonic graphs in the Heisenberg group and their properties.

We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing …

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the …

Tensor measures chirality for curves, even those with rough edges.

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

A directed graph $G$ is $\textit{intrinsically linked}$ if every embedding of that graph contains a non-split link $L$, where each component of $L$ is a consistently oriented cycle in $G$. A $\textit{tournament}$ is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intr…

Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.

We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several ex…

Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of $\tria…

We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.

Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrin…

The paper proves Rademacher's theorem for Heisenberg groups.

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipa…

Criterion for surfaces in Heisenberg group to be graphs using flat cones.

Ambitwistor string matches superstring chiral integrands at zero tension.