Geometric Techniques in Knot Theory – Jessica S. Purcell
Jessica S. Purcell – Brigham Young University; von Neumann Fellow, School of Mathematics
October 20, 2015, Institute for Advanced Study
Techniques are referred to as machinery 0.50
She’s been asked to present this because it’s useful for somebody else’s talk.
She begins by outlining the structure of the talk, how it’s going to go down.
Manifolds are not complements
“Decompositions of knots/link complements into polyhedra”
1. chop complicated manifold into extremely simple pieces. “just balls with extra information”
2. Put more structure on the polyhedra – angle structures. Look at surfaces that sit inside (Put more structure on? Like, more carefully defined?)
3. examples and applications throughout 3.04
1. k in s3 – cut this into polyhedron – 3-ball with extra info on boundary.
1. Each vertex valence >> three. No ___.___
2. Each edge has end points on distinct vertices. No O.
“I guess what I’m trying to say here is that sometimes a polyhedron is just a graph on the boundary of a 3-ball but we’re gonna restrict to the graphs with these two properties”
polyhedron is just a graph on the boundary of a 3-ball?
5.13 first decomposition into ideal polyhedra. Due to MENASCO THURSTON
6.00 The knot or link sits on projection plane. K sitting on s2 (notation)
Thicken up the plane – goes up and down but stays inside the plane (!!)
Put a ball above and a ball below.
6.50 Consults with audience – “You need a little more info on how to glue this together”
- Balloons meet in faces corresponding to regions of the diagram.
- You get an edge where faces meet. Edges are often called crossing arcs.
- Vertices are remnants of the knot. Overstrands top, understrands bottom.
9.23 another example to play/work with
Asks the audience: do you understand?
10.01 Second decomposition due to AGOL THURSTON. Fully augmented link
Two types of components.
1. Lie entirely → embedded. “sit in here”, in the plane, OR single crossing
2. orthog to plane, an unknotted component – crossing 2
1. “first thing you’re gonna do…” Each crossing circle bounds a crossing disk. “these are gonna give me shaded faces”. Cut along it. A full slice of pitta bread sliced down the middle – you can undo single crossings (Rotate!) → all flat in the projection plane! Why is it always food? Because it’s like a recipe? (Link complement you can put in as many twists as you like…???? 15.24)
2. Slice along projection plane. Two polyhedra – sketches DIAGRAMS
3. Flatten. Crossing circles become (sketch)
4. Collapse to ideal vertices. ‘white faces’
cut and unwind shrink to ideal vertices
22.16 exactly the same despite half twists. If … glue to the guy in the back. This part… I don’t understand the polyhedron bit
Angle structures – “I need to be able to talk about truncated vertices”.
Truncate those that are going off to infinity, and create a face. Leftovers are called interior edges.
25.52 A normal disk inside the polyhedron boundary is a normal curve. That’s “kind of a useless definition”. A normal curve sits inside a boundary and satisfies these conditions:
1. transverse to edges.
“think of these as things you’ll do to your disk if it’s not in a good position. If it’s not transverse to edges just bump it a little and it will be.” 26.33
2. does not lie in a single face. “push it through”
3. No arc of gamma has end pts on same edge, or interior edge and adjacent edge. “slide that past”
4. Meets each edge at most once and each boundary face at most once
“now we’re ready to talk about the definition of an angled polyhedral decomposition” 28.50
subdivisions of link component into ideal polyhedra – truncate them
such that each interior edge is assigned an angle between 0 and pi and an exterior angle (“really just for notational purposes”)
these must satisfy
1. if you sum up all the angles around an edge you get 2 pi
2. if gamma is a normal curve meeting only interior edges then …. if gamma encircles an ideal vertex.
32.04 definition of an angle structure.
dots are truncated – boundary cases. Assign all edges to the angle
“maybe the right picture to draw is this one” 34.00
1. all edges 4-valent 34.30
2. check that meeting 4 edges
“you’re gonna wanna use a knot that’s prime” 36.30
37.34 how this is normally used. “End up with something hyperbolic”.
Combinatorial area of disk d. 38.30. Talk about normal disk, generalise to more complicated disks.
Interior edges met by boundary of d.
“define the combinatorial area as sum of ext angles plus pi times number of intersections with boundary faces”40.00
the intuition this comes from hyperbolic geometry
exactly the formula for hyp. Area of polygon.
Couple of examples
1. 2 special types of faces. So I have my polyhedron doing something. Here is a disk in orange. It’s meeting 2 boundary faces and no interior edges so the area of this guy is going to be zero plus two times pi – 2 pi 0. This is called a boundary bigon.
2. Another example would e a disk that runs parallel to a boundary face so a vertex link also area zero by definition of an angled structure. 43.00
“Looks like I want to extend the definition to what are called admissible disks”. 43.56
Similar to normal disks, but allow them to be immersed – no longer require that are embedded. Boundary is a curve (REFERS BACK TO DEF. OF NORMAL) 45.02
Admissible surface meets each polyhedron in admissible disks.
46.00 “let me give you a reason to think that this might be something useful to study”
47.38 length of gamma is area divided by number of arcs on boundary faces
Stuff about lengths of slopes – all pretty technical.
“If you have a manifold with angled polyhedral decomposition and a collection of slopes then manifold obtained by DEHN filling is hyperbolic”.
Theorem by Futer and I says that if least 6 crossings per twist region then your knot complement is going to be hyperbolic.
Twist region looks like this 53.30. Untwist.
Go from one to the other – do a DEHN filling on cross circle.
Knot → shell → flatten → triangles. Disk inside the polyhedron; combinatorial area = sum of ext. angles + pi x no. of intersections with boundary faces.