Abstract: Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is realizable as the Gauss code of an immersed graph.
Abstract: We show that, given any $n$ and $\alpha$, every embedding of any sufficiently large complete graph in $\mathbb{R}^3$ contains an oriented link with components $Q_1$, ..., $Q_n$ such that for every $i\not =j$, $|\lk(Q_i,Q_j)|\geq\alpha$ and $|a_2(Q_i)|\geq\alpha$, where $a_{2}(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$.
Abstract: We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including $K_7$, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.
Abstract: Two natural generalizations of knot theory are the study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spatial graphs.
Abstract: In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).
Abstract: We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.
Abstract: Beaded beads are clusters of beads woven together (usually around one or more large holes). Their groups of symmetries are classified by the three-dimensional finite point groups, i.e. the finite subgroups of the orthogonal group of degree three, O(3). The question we answer is whether every finite subgroup of O(3) can be realized as the group of symmetries of a beaded bead. We show that this is possible, and we describe general weaving techniques we used to accomplish this feat, as well as examples of a beaded bead realizing each finite subgroup of O(3) or, in the case of the seven infinite classes of finite subgroups, at least one representative beaded bead for each class.
Abstract: Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results. We state the definitions, provide some examples, and survey the known results. We hope that this paper will help lead to rapid development of the area.
Abstract: 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 examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.
Abstract: In previous work, the author defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.
Abstract: We prove that a graph is intrinsically linked in an arbitrary 3-manifold M if and only if it is intrinsically linked in S3. Also, assuming the Poincare Conjecture, a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S3.
Abstract: We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.
Abstract: We derive formulas for counting the number of strands in a variety of knotwork designs inspired by traditional Celtic designs, including rectangular panels, circular borders, rectangular borders, and half frames.
Abstract: We give a geometric interpretation of Milnor's invariants $\bar{\mu}(ijk)$ in terms of triple intersection points of Seifert surfaces for the three link components. This generalizes ideas of Cochran to links which are not algebraically split.
Abstract: We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides, and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of marked chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.
Abstract: We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author.
Abstract: We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.
Abstract: This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components. (Note: This main result later turned out to be false - see "On the existence of finite type link homotopy invariants")
Abstract: Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams."
Abstract: Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components. (Note: This main result later turned out to be false - see "On the existence of finite type link homotopy invariants")
For more papers on finite type invariants, check out the bibliography maintained by Sergei Duzhin.
Versions of my papers (and many others) are available at the Front for the Mathematics ArXiv.
Return to Blake's Homepage.