arXiv research

New knot graphs show most are not Gromov hyperbolic, with special cases.

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

Dominant knots have isomorphic Seifert and Tait graphs.

New signatures for knotted graphs linked to classical knot signatures.

Graphs from knot types help identify unique knots.

Extends knot polynomial to knotted 4-valent graphs.

Two proofs show Kinoshita graph is knotted despite simple edge removals.

Survey of intrinsically linked or knotted graphs.

Most graphs are knotted as they grow larger.

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…

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…

A new invariant for knotted graphs defined by label bracket.

Newly discovered 5 triangle-free intrinsically knotted graphs with 22 edges.

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.

Graphs and their complements are intrinsically knotted.

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

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 $K_7$…

New invariant for surface-knots in 4D from marked graphs.

Geometric deep learning predicts knot invariants.

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…

New method finds knots without low treewidth diagrams.

In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with this correspondence. This correspondence is given by a simple formula.

Proof that critical knots of Morse-Bott functions are graph knots.

Virtual knots with same writhe polynomial have equivalent intersection graphs.

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…

Graphs represent knot adjacency for n crossings.

Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.

The paper generalizes virtual knot theory using multiple types of virtual crossings.

New algebraic structures biquasiles defined using dual graph diagrams for knot and link invariants.

Directed graphs can be intrinsically knotted and 4-linked.

Study graph manifolds, knots, and mapping classes via profinite groups.

In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ …

Quantum model for knotted graphs from knot theory.

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precros…

Study on spatial graphs and their constituent knots, linking polynomial invariants.

Geometric duality connects graph isomorphism and knot equivalence.

We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…

Researchers developed an algorithm to count all graph mosaics.

Extends knot concordance invariant to balanced spatial graphs using grid homology.

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization …

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

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in $S^3$.

This paper is an exploration of simple four-regular graphs in the plane (i.e. loopless and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921 -- 2005) whose fa…

The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.

New knot invariant from 3-braids and 6-valent graphs.