IlliTantrixA new way of looking at knot projections

Yana Malysheva, Amit ChatwaniIlliMath2002

Mentors•Elizabeth Denne

– Principal Mentor

•Prof. John Sullivan– Corresponding Mentor

•Prof. George Francis– Director IlliMath 2002

Funding Provided By:• NSF VIGRE

– National Science Foundation – Vertical Integration Research in Education

• UIUC MATHEMATICS DEPARTMENT– University of Illinois at Urbana Champaign

• NCSA– National Center for Super Computer Applications

Definition of a Knot

• A knot is a simple closed curve K in R3, such that K is homeomorphic to a circle.

• IlliTantrix works with stick knots – knots composed of a finite number of sticks.

Smooth knot

Stick knot

Examples:

Projections of a knot

• A knot in R3 can be projected onto a plane.

• Different projections of the same knot may have a different number of crossings (places where the projection intersects itself.)

Two projections of a trefoil knot

3 crossings

6 crossings

Regular projections

• We are interested in regions of regularity – those projections in which you can definitely count the number of crossings.

• These regions will be separated by curves of irregular projections.

Examples of irregular projections:

trisecants

overlapping edges

vertices on edges

Crossing map

• The crossing map of a knot captures the change in the number of crossings you see as you change your view of the knot.

• A point on the sphere corresponds to a direction in which to view the knot. This view will have a number of crossings. The crossing map assigns each point on the sphere this number.

Aim of the project

The aim of this project is to visualize the crossing map of a knot in a real-time interactive computer animator (RTICA).

It was inspired by the work of Colin Adams.

Features of the crossing map

• Moving across 1-curves, the number of crossings changes by one.

1 - curves

Change of view across a 1-curve:

this is where the 1-curve is

• The tantrix (tangent indicatrix) is the curve of directions of unit tangent vectors of the knot.

• For stick knots, this is the arc of the great circle connecting two consecutive directions.

• When looking in a tangent direction, you will see part of a 1-curve.

Tantrix

Two edges of the knot and the corresponding part of the 1-curve

Features of the crossing map

• Moving across 2-curves, the number of crossings changes by two.

2-curves

change of view across the 2-curve:

this is where the 2-curve is

Constructing the 2-curve

• The two edges adjacent to the vertex v lie on the same side of the plane spanned by v and edge e.

• The 2-curve is the arc of the great circle connecting the two vectors from v to the endpoints of e.

v

e

Trisecant Curves

• A trisecant is a triple of collinear points of the knot.

• The trisecant curve captures the directions in which you see a trisecant.

• Moving across trisecant curves does not change the number of crossings. this is where the trisecant is

Trisecant Curves

• We care about trisecant curves because we know that when a trisecant curve intersects itself, there is a quadrisecant.

• Since we know that every knot has a quadrisecant, we also know that every knot has a projection with a least six crossings.

Changing our view from a quadrisecant, we see six crossings:

Vertex-Eye View curves

• Vertex-Eye View curves are curves on the crossing map that represent all the directions in which you would look from a specified vertex, V , and see a part of the knot.

• Parts of the VEV curve correspond to parts of the 1, 2 and trisecant curves.

Vi

i

When curves meet

• Curves often meet and intersect each other on the crossing map. When that happens, we can predict the change in the number of crossings in the adjacent regions.

1

1

k

k+1

k+1

k+2

If two 1-curves intersect:

1

2

k

k+1k+3

k+2

If a 1-curve and a 2-curve intersect:

1

2

kk+1

k+2

If a 2-curve meets a 1-curve:

When curves meet

• In some situations, there are two regions whose number of crossings differs by 4.

• We also know that for any knot that is not an unknot, the minimum number of crossings in any projection is 3.

If two 2-curves intersect:

2

2

k

k+2k+4

k+2If a 2-curve is

intersected by two 1-curves going in the

same direction:

2

1 1

k

k+2 k+4

k+2

k+3

k+1

If we could prove that any trefoil knot’s crossing map contains at least one of these cases, then we would know that every trefoil has a projection with at least 7 crossings, a conjecture knot theorists have been trying to prove.

Future developments

• The trisecant curve requires a lot of calculation to derive. For that reason, it is not currently calculated in IlliTantrix.

• One of the future changes could be to add the trisecant curve to the program.

Trisecant curve

Future developments

• A point and its antipode have the same number of crossings.Thus, the crossing map is actually a map from RP Z .

• In the future, one could change the visualization of the crossing map to represent that.

A more accurate visualization of the crossing map

2 +