labeled oriented intervals that are not...

Labeled Oriented Intervals that are not

Diagrammatically Reducible

Ashley Earls

St. Olaf College

Northfield Undergraduate Mathematics Symposium, September25, 2012

Collaborative Work

This research was a collaborative effort with Dr. Jens Harlander(mentor), Rachael Keller, Gabriel Islambouli, and Mingjia Yang.

Our work took place at Boise State University this past summer.

Ashley Earls Labeled Oriented Intervals that are not Diagrammatically Reducible

What is a virtual knot?

A knot is a 4-regular planar graph with over- and under-crossinginformation provided at the vertices.

A virtual knot is simply a knot that is not necessarily planar. Aknot can always be embedded in a plane or a sphere; however, avirtual knot sometimes lies on a surface with holes, such as a torus.

LOGs and LOIs

A labeled oriented graph (LOG) is a directed graph on vertices{1, . . . , n} where edges are labeled by vertices.

A labeled oriented tree (LOT) is a LOG whose underlying graph isa tree, and a labeled oriented interval (LOI) is a LOG whoseunderlying graph is an interval.

Our research deals primarily with LOIs.

LOGs and LOIs

Our research deals primarily with LOIs.Ashley Earls Labeled Oriented Intervals that are not Diagrammatically Reducible

2-cells

A 2-cell is a polygon whose edges are labeled and oriented. Theinside of a 2-cell can be considered an open disk.

Each edge of a LOT encodes a 2-cell:

2-cells

Each 2-cell can then be dualized to look like a crossing on a knot.

So, each LOT corresponds to a knot (sometimes virtual). Eachknot therefore has a set of “tiles” given by its crossings.

2-cells

2-complexes

A 2-complex is obtained by creating a set of 1-cells (open arcs)which intersect in a single point, and gluing the 2-cells into thisobject while respecting labels and orientations.

For example, a trefoil complex is given by:

2-complexes

A 2-complex is obtained by creating a set of 1-cells (open arcs)which intersect in a single point, and gluing the 2-cells into thisobject while respecting labels and orientations.

For example, a trefoil complex is given by:

Spherical Diagrams

Specifically, we are concerned with constructing sphericaldiagrams. These are surfaces constructed by gluing together 2-cellswhich can then be deformed into a sphere.

A spherical diagram which uses the tiles of a LOT can also bethought of as a mapping from the 2-sphere into the LOT complex.

Spherical Diagrams

Reducing Spherical Diagrams

Given a spherical diagram, it is sometimes possible to reduce thenumber of tiles used by cutting and reorganizing them. There aretwo ways to do this.

Edge fold:

Given a spherical diagram, it is sometimes possible to reduce thenumber of tiles used by cutting and reorganizing them. There aretwo ways to do this.

Edge fold:

Vertex fold:

If an edge or vertex fold is possible with a given spherical diagram,then performing this reduction replaces the initial mapping into theLOT complex by a new, simpler one.

Vertex fold:

If an edge or vertex fold is possible with a given spherical diagram,then performing this reduction replaces the initial mapping into theLOT complex by a new, simpler one.

A spherical diagram is called reduced if no edge folds or vertexfolds are possible.

A LOT is called diagrammatically reducible (DR) if it admits noreduced diagrams (i.e., every spherical diagram built from the2-cells of the LOT has a possible vertex fold or edge fold).

A spherical diagram is called reduced if no edge folds or vertexfolds are possible.

A LOT is called diagrammatically reducible (DR) if it admits noreduced diagrams (i.e., every spherical diagram built from the2-cells of the LOT has a possible vertex fold or edge fold).

Why do we care?

Whitehead’s conjecture (1941) states that a sub-complex of anaspherical 2-complex is aspherical. This remains an open question.

A 2-complex is aspherical if processes such as edge and vertexfolding can eventually reduce it to the empty spherical map.

In short,

Whitehead’s conjecture essentially says that all LOTcomplexes are aspherical

DR ⇒ aspherical

We therefore look for and examine LOTS which are non-DR

Why do we care?

In short,

DR ⇒ aspherical

Why do we care?

In short,

DR ⇒ aspherical

The Rosebrock LOTs

Three years ago, Dr. Stephan Rosebrock, a researcher at thePadagogische Hochschule in Karlsruhe, Germany, began toinvestigate diagrammatic reducibility for LOIs using computers.

The program checked billions of LOIs and found only few reducedspherical diagrams.

Main goals of our research:

Find common properties of these LOIs and their diagrams

Generalize findings into constructions which produce reduceddiagrams

Investigate whether these diagrams are aspherical

All the diagrams we will look at today came from Stephan’scomputer.

The Rosebrock LOTs

erz10a5

This diagram is reduced - no crossing is adjacent to its mirrorimage.

This LOI is one of the smallest possible to give a reduced diagram.(We have proven that there can be no reduced diagram from a LOIwith four or fewer vertices.)

The “Orange Peel” Construction

Theorem

For any even n ≥ 6,

has a reduced diagram when k = n2.

When n = 6, the LOI from this construction is:

The reduced diagram comes directly from the LOI for thisconstruction. First dualize and then connect adjacent crossings:

Note that there are two unconnected strands. This means ourspherical diagram is incomplete.

This section of the diagram goes in as 6 and comes out 1

Will be referred to as “orange” in upcoming slides

Note that there are two unconnected strands. This means ourspherical diagram is incomplete.

This section of the diagram goes in as 6 and comes out 1

Will be referred to as “orange” in upcoming slides

In a complete spherical diagram, each crossing has a mirror imagesomewhere else in the tiling. So this time, we flip the crossingsfirst.

Again, connect adjacent tiles.

Begin as before:

The “Orange Peel Construction

We can insert another copy of the orange to fill in the gaps:

Why is this a reduced diagram?

Oranges are not connected at corresponding ends

Inverse edges act as a buffer

We have seen the construction works when n = 6, and it is possibleto prove it also works for all larger even n by using induction.

Acknowledgements

Thanks to NSF for funding our research under grant DMS1062857, Boise State University for hosting the REU, and Dr.Stephan Rosebrock for the data and continual help.

References

W.A. Bogley, J.H.C. Whitehead’s asphericity question, in“Two-dimensional Homotopy and Combinatorial GroupTheory”, edited by C. Hog-Angeloni, W. Metzler, A.J.Sieradski, LMS Lecture Note Series 197, CUP 1993.

J. Harlander, S. Rosebrock, Generalized knot complements andsome aspherical ribbon disc complements, Journal of KnotTheory and Its Ramifications, Vol 12, No. 7 (2003) 947-962.

J. Howie, On the asphericity of ribbon disc complements,Trans A.M.S. 289(1) (1985) 281-302.

G. Huck, S. Rosebrock, Aspherical labeled oriented trees andknots Proc. of the Edinburgh Math. Society (44) (2001)285-294.

S. Rosebrock, A reduced spherical diagram into a ribbon–diskcomplement and related examples, Editor: P. Latiolais, Lecturenotes in mathematics 1440, Springer Verlag (1990), 175-185.

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