MT855: Combinatorial Methods in Knot Theory
February 25, 2013
13
Coverings and colorings, part II
We continue to explore the relationship between knot colorings, dihedral representations of the knot
group, and characters on the rst homology of the branche
MT855: Combinatorial Methods in Knot Theory
February 4, 2013
8
Lens spaces and the fundamental group
Today we examine the lens spaces in a bit greater detail, determine their fundamental groups, and
then segue into a discussion about the knot group.
We as
MT855: Combinatorial Methods in Knot Theory
February 4, 2013
8
Lens spaces and the fundamental group
Today we examine the lens spaces in a bit greater detail, determine their fundamental groups, and
then segue into a discussion about the knot group.
We as
MT855: Combinatorial Methods in Knot Theory
February 13, 2013
10
The knot group, part II
Today we look at the relation of the knot group to knot colorings and to the fundamental group of
Dehn surgery along a knot.
Last class we distinguished the unknot, r
MT855: Combinatorial Methods in Knot Theory
February 20, 2013
12
Coverings and colorings, part I
Deep results to come will show that the homeomorphism type of XK is a complete invariant of the
knot type K and that (K ) is nearly so (it is with its periphe
MT855: Combinatorial Methods in Knot Theory
February 18, 2013
11
11.1
Seifert surfaces and the fundamental group of a Dehn surgery
Seifert surfaces.
Lets make sure we can establish the following foundational fact that appeared on the rst homework.
Proposi
MT855: Combinatorial Methods in Knot Theory
February 1, 2013
7
Dehn surgery, part III
We begin with some hints on the rst homework. You were asked to show that if K R3 bounds a
smoothly embedded disk h : D R3 , then K is isotopic to a round unknot. The id
MT855: Combinatorial Methods in Knot Theory
January 30, 2013
6
Dehn surgery, continued
We have a shortage of knots at the moment. To address this, let U
S 3 denote the unknot. Then
the curve of slope p q on XU
S 3 gives a knot in S 3 . By denition, this i
MT855: Combinatorial methods in knot theory
Professor Josh Greene
Boston College
January 16, 2013
1
Foundations
What is a knot? Here is a sensible rst denition.
Denition 1.1 (Topological Knot) A knot is a continuous, one-to-one map : S 1 S 3 .
When should
MT855: Combinatorial Methods in Knot Theory
January 18, 2013
2
The Mazur Swindle
Wild knots help us probe the distinction between continuous and smooth topology in three dimensions,
and they can even be used to tell us something interesting about smooth k
MT855: Combinatorial Methods in Knot Theory
January 23, 2013
3
Homology and curves on surfaces, Part I
We begin with the calculation of the ordinary homology of a knot complement S 3 \ K .
Proposition 3.1 If K S 3 is a knot, then
H ( S 3 \ K ) =
Z,
0,
=
MT855: Combinatorial Methods in Knot Theory
January 28, 2013
5
Curves on the torus, the Poincar conjecture, and Dehn surgery
e
We are applying what we know about curves on surfaces to the particular case of T 2 . Recall the
Proposition stated at the end o
MT855: Combinatorial Methods in Knot Theory
January 25, 2013
4
Curves on surfaces, part II
Recall that were on course to prove the bigon criterion. Picking up from our last lecture, we hit a
minor obstacle in locating a bigon during the proof. The relevan