Complete bipartite graphs whose topological symmetry groups are polyhedral

We determine for which $n$, the complete bipartite graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.Comment: 25 pages, 6 figures, latest version has minor edits in preparation for submissio

A few weight systems arising from intersection graphs

We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], 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 {\it 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.Comment: 20 pages. This version has been substantially revised. The results are largely the same, but the proofs have been reconceptualized in terms of various 2-term relations on chord diagrams and graph

Alexander and writhe polynomials for virtual knots

We give a new interpretation of the Alexander polynomial $\Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $\Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.Comment: 22 pages, 19 figure

The Intersection Graph Conjecture for Loop Diagrams

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."Comment: 23 pages, many figures. arXiv admin note: Figures 1, 2, 5 and 11 included in sources but in format not supported by arXi

Tree Diagrams for String Links II: Determining Chord Diagrams

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).Comment: 14 pages, many figure

Finite Type Link Homotopy Invariants

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.Comment: 15 pages, many figures. Revised to acknowledge work of Bar-Natan, Garoufalides, Rozansky and Thurston. Revised again to clarify the exposition in section

Tree Diagrams for String Links

In previous work, the author defined the intersection graph of a chord diagram associated with string links (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.Comment: 12 pages, 14 figure