3,298 research outputs found
On the tree-width of knot diagrams
We show that a small tree-decomposition of a knot diagram induces a small
sphere-decomposition of the corresponding knot. This, in turn, implies that the
knot admits a small essential planar meridional surface or a small bridge
sphere. We use this to give the first examples of knots where any diagram has
high tree-width. This answers a question of Burton and of Makowsky and
Mari\~no.Comment: 14 pages, 6 figures. V2: Minor updates to expositio
A Structural Approach to Tree Decompositions of Knots and Spatial Graphs
Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion
On knot Floer width and Turaev genus
To each knot one can associated its knot Floer homology
, a finitely generated bigraded abelian group. In general, the
nonzero ranks of these homology groups lie on a finite number of slope one
lines with respect to the bigrading. The width of the homology is, in essence,
the largest horizontal distance between two such lines. Also, for each diagram
of there is an associated Turaev surface, and the Turaev genus is the
minimum genus of all Turaev surfaces for . We show that the width of knot
Floer homology is bounded by Turaev genus plus one. Skein relations for genus
of the Turaev surface and width of a complex that generates knot Floer homology
are given.Comment: 15 pages, 15 figure
A Turaev surface approach to Khovanov homology
We introduce Khovanov homology for ribbon graphs and show that the Khovanov
homology of a certain ribbon graph embedded on the Turaev surface of a link is
isomorphic to the Khovanov homology of the link (after a grading shift). We
also present a spanning quasi-tree model for the Khovanov homology of a ribbon
graph.Comment: 30 pages, 18 figures, added sections on virtual links and
Reidemeister move
The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams
This work presents formulas for the Kauffman bracket and Jones polynomials of
3-bridge knots using the structure of Chebyshev knots and their billiard table
diagrams. In particular, these give far fewer terms than in the Skein relation
expansion. The subject is introduced by considering the easier case of 2-bridge
knots, where some geometric interpretation is provided, as well, via
combinatorial tiling problems.Comment: 20 pages, 4 figures, 2 table
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