Knotted Graphs Research Articles

Planar Stick Indices of Some Knotted Graphs

Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.

Knotted 4-regular graphs. II. Consistent application of the Pachner moves

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.

Hyperbolic 3-manifolds with Boundary Which are Side-pairings of Two Tetrahedra as Exteriors of Knotted Graphs in the 3-sphere

In this paper, we give a generalization of Ivanšić’s method for hyperbolic 3-manifolds without boundary, which allows us to recognize if a hyperbolic 3-manifold with totally geodesic boundary, given by an isometric side-pairing of two hyperbolic truncated tetrahedra, is the exterior of a knotted graph; i.e., it is the complement of a 1-manifold with isolated singularities embedded in S3, in which case we get the corresponding diagram of the knotted isotopy class of its boundary. Otherwise, we obtain that the corresponding 3-manifold with boundary is the exterior of a knotted graph embedded in some lens space. Finally, we apply this method to a noncompact 3-manifold with a totally geodesic surface boundary of genus 2.

Intrinsically knotted graphs with linklessly embeddable simple minors

Intrinsically knotted graphs with linklessly embeddable simple minors

Bipartite intrinsically knotted graphs with 23 edges

A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the E9+e family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex has degree 3 or more. Four among them contain the Heawood graph and the other two contain Cousin 110 of the E9+e family. Consequently, there is no minor minimal intrinsically knotted graph with 23 edges that is bipartite.

Knotted 4-regular graphs: Polynomial invariants and the Pachner moves

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs—one inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.

Quotients of the Gordian and H(2)-Gordian graphs

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

Coloring in graphs of twist knots

AbstractLet Tn be a twist knot with n half‐twists and Gn be the graph of Tn. The closed neighborhood N[v] of a vertex v in Gn, which included at least one colored vertex for each color in a proper n‐coloring of Gn, is called a rainbow neighborhood. There are different types of graph coloring in the literature. We consider some of these types in here. In this paper, we determine the chromatic number of graphs of twist knots and study rainbow neighborhood of graphs of twist knots. We determine the rainbow neighborhood number and the fading number of them. Furthermore, we determine coupon coloring and the coupon coloring number of graphs of twist knots.

More intrinsically knotted graphs with 22 edges and the restoring method

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 [Formula: see text] and the 13 graphs obtained from [Formula: see text] by [Formula: see text] moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper, we show that there are exactly five triangle-free intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the [Formula: see text] family, Cousins 97 and 99 of the [Formula: see text] family and two others that were previously unknown.

A new intrinsically knotted graph with 22 edges

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) proved there are exactly 14 intrinsically knotted graphs with 21 edges by showing that H12 and C14 are the only triangle-free intrinsically knotted graphs of size 21. Our current goal is to find the complete set of intrinsically knotted graphs with 22 edges. To this end, using the main argument in [9], we seek triangle-free intrinsically knotted graphs. In this paper we present a new intrinsically knotted graph with 22 edges, called M11. We also show that there are exactly three triangle-free intrinsically knotted graphs of size 22 among graphs having at least two vertices with degree 5: cousins 94 and 110 of the E9+e family, and M11. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5.

Singular links and Yang-Baxter state models

We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Speci\-fically, we consider a Yang-Baxter state model for the {\rm sl($n$)} polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra and seek conditions for further extending the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the {\rm sl($n$)} polynomial for classical links.

Graphs With Disjoint Links In Arbitrary 3-Manifold

In recently years, intrinsically linked and intrinsically knotted graphs, as very important graphs in spatial graphs are becoming targets of some relatively new research areas. This paper is devoted to the graph disjoint linking in is disjoint linked in arbitrary 3-manifold,and show that the graph is minor minimal with respect to intrinsically disjoint linked graph in arbitrary 3-manifold. The theorem is proved by the lemmas, the conclusion is concluded.

Bipartite Intrinsically Knotted Graphs with 22 Edges

AbstractA graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, K7 and the 13 graphs obtained from K7 by moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1, 9, 11, 12]. This set includes exactly one bipartite graph, the Heawood graph. In this article we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the K3, 3, 1, 1 and families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.

Graphs on 21 edges that are not 2-apex

We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on $K_7$, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not $2$--apex. As a corollary, this gives a new proof that the 14 graphs obtained by triangle-Y moves on $K_7$ are the minor minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven graph Petersen family, obtained from $K_6$, is the set of graphs of at most 17 edges that are minor minimal for the property not apex.

Exactly fourteen intrinsically knotted graphs have 21 edges

Throughout the article we will take an embedded graph to mean a graph embedded in R . We call a graph G intrinsically knotted if every embedding of the graph contains a knotted cycle. Conway and Gordon [2] showed that K7 , the complete graph with seven vertices, is an intrinsically knotted graph. A graph H is minor of another graph G if it can be obtained from G by contracting or deleting some edges. An intrinsically knotted graph is minor minimal intrinsically knotted provided no proper minor is intrinsically knotted. Robertson and Seymour [9] proved that there are only finite minor minimal intrinsically knotted graphs, but finding the complete set of them is still an open problem. However, it is well known that K7 and the thirteen graphs obtained from this graph by rY moves are minor minimal intrinsically knotted; see Conway and Gordon [2], and Kohara and Suzuki [6].

Twist spinning knotted trivalent graphs

In 1965, E. C. Zeeman proved that the ( ± 1 ) (\pm 1) -twist spin of any knotted sphere in ( n − 1 ) (n-1) -space is unknotted in the n 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 2 -dimensional foam which is a generalization of a knotted sphere and prove that a ( ± 1 ) (\pm 1) -twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the ( ± 1 ) (\pm 1) -twist spins are always unknotted.

There exist no minimally knotted planar spatial graphs on the torus

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.

Excluded Minors and the Ribbon Graphs of Knots

AbstractIn this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.

Deleting an edge of a 3-cycle in an intrinsically knotted graph gives an intrinsically linked graph

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

Intrinsically Knotted and 3-Linked Graph

Intrinsically linked graphs are very important spatial graphs. We say that is intrinsically knotted and 3-linked graph if every spatial embedding of this graph contains nontrivial knot and a non-split 3-component link. This paper exhibit a new intrinsically knotted and 3-linked graph. This paper is devoted to the operation of the exchanges preserve intrinsically knotted and 3-linked graph.