Daniel Irvine August 1, 2013 SMALL 2013dirvine/Documents/Presentations/DI_Uber... · Overview The...

OverviewThe Ubercrossing Algorithm

Ubercrossing NumbersConclusion

Ubercrossing Projections of Knots

Daniel Irvine

August 1, 2013

SMALL 2013

Daniel Irvine Ubercrossing Projections of Knots

Background

Traditionally, knot theorists have studied knots by considering knotprojections having crossings where only two strands intersect at atime. Let’s call this kind of crossing a double crossing.

Earlier in the day, we were introduced to a new way to thinkabout the crossings in a knot.

Background

Instead of restricting our attention to only double crossings, let’sconsider crossings where three, four, or more strands are allowed tointersect at each crossing. These kinds of crossings are calledtriple crossings, quadruple crossings, etc, and they are a new wayto measure the complexity of a knot.

We defined n-crossing to be a point in a knot projection where nstrands cross.

Background

Instead of restricting our attention to only double crossings, let’sconsider crossings where three, four, or more strands are allowed tointersect at each crossing. These kinds of crossings are calledtriple crossings, quadruple crossings, etc, and they are a new wayto measure the complexity of a knot.

We defined n-crossing to be a point in a knot projection where nstrands cross.

What is an Ubercrossing?

Question

Can we find a projection in which a knot has only a singlen-crossing?

If this is possible, we will call such a crossing an ubercrossing, andwe will call such a projection an ubercrossing projection.

Question

The Ubercrossing Algorithm

Theorem (SMALL 2012)

Every knot has an ubercrossing projection.

Proof: We present the ubercrossing algorithm, which will showhow to obtain an ubercrossing projection for any knot.

Theorem (SMALL 2012)

Every knot has an ubercrossing projection.

Proof: We present the ubercrossing algorithm, which will showhow to obtain an ubercrossing projection for any knot.

Ubercrossing Numbers

The ubercrossing algorithm gives an ubercrossing projection of anyknot. This ubercrossing projection is certainly not unique.

Definition

The ubercrossing number of a knot K , denoted u(K), is the leastnumber of strands in any ubercrossing projection of K ; that is, theleast n such that the n-crossing number of K is 1.

Ubercrossing Numbers

The ubercrossing algorithm gives an ubercrossing projection of anyknot. This ubercrossing projection is certainly not unique.

Definition

The ubercrossing number of a knot K , denoted u(K), is the leastnumber of strands in any ubercrossing projection of K ; that is, theleast n such that the n-crossing number of K is 1.

Finding u(K)

Finding the ubercrossing number for a knot can be quite difficult.

Instead, we begin with a single n-crossing and connect the strandsin all possible ways. From this we can generate a list of knots withubercrossing number less than or equal to n.

Finding u(K)

Finding the ubercrossing number for a knot can be quite difficult.Instead, we begin with a single n-crossing and connect the strandsin all possible ways. From this we can generate a list of knots withubercrossing number less than or equal to n.

Finding u(K)

Finding the ubercrossing number for a knot can be quite difficult.Instead, we begin with a single n-crossing and connect the strandsin all possible ways. From this we can generate a list of knots withubercrossing number less than or equal to n.

K u(K) K u(K) K u(K) K u(K)

31 4 81 7 815 8 31#31 641 5 82 8 816 8 31#31 651 6 83 7 817 8 31#41 752 6 84 7 818 ? 31#51 861 6 85 8 819 6 31#51 762 6 86 7 820 6 31#52 863 6 87 7 821 6 31#52 771 7 88 7 91 8 41#41 872 7 89 7 92 873 7 810 8 93 874 7 811 8 94 875 7 812 8 95 876 7 813 8 96 877 7 814 8 97 8

Questions and Conjectures

Can we find the ubercrossing number for an infinite family ofknots, such as (p, q)-torus knots?

For (r , r + 1) torus knots, u = 2r .

How does ubercrossing number behave under composition?

Acknowledgements

SMALL 2013 Knot Theory GroupProfessor Colin Adams

Orsola Capovilla-Searle, Jesse Freeman, Daniel Irvine, SamanthaPetti, Daniel Vitek, Ashley Weber, Sicong “Scott Chaos” Zhang.

References

Colin Adams, Thomas Crawford, Benjamin Demeo, MichaelLandry, Alex Tong Lin, MurphyKate Montee, Seojung Park,Saraswathi Venkatesh, Farrah YheeKnot Projections with a Single Multi-Crossing. 2012.

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