136 research outputs found
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
Complete bipartite graphs whose topological symmetry groups are polyhedral
We determine for which , the complete bipartite graph has an
embedding in whose topological symmetry group is isomorphic to one of the
polyhedral groups: , , or .Comment: 25 pages, 6 figures, latest version has minor edits in preparation
for submissio
Alexander and writhe polynomials for virtual knots
We give a new interpretation of the Alexander polynomial for
virtual knots due to Sawollek and Silver and Williams, and use it to show that,
for any virtual knot, 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
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