1. IlliTantrix A new way of looking at knot projections Yana Malysheva, Amit Chatwani IlliMath2002

2. Mentors Elizabeth Denne – Principal Mentor Prof. John Sullivan – Corresponding Mentor Prof. George Francis – Director IlliMath 2002

3. 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

4. Definition of a Knot A knot is a simple closed curve K in R 3, 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:

5. Projections of a knot A knot in R 3 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

6. 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

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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:

15. 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. V i i

16. 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+2 If two 1-curves intersect: 1 2 k k+1 k+3 k+2 If a 1-curve and a 2-curve intersect: 1 2 k k+1 k+2 If a 2-curve meets a 1-curve:

17. 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+2 k+4 k+2 If a 2-curve is intersected by two 1- curves going in the same direction: 2 11 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.

18. 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

19. 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 +