Sections 1.1 and 1.2
Read 1.1 and 1.2
-
Composition, Prime Knots
Continue on homework
Def: A composite knot (connected sum)is the
composition J#K of nontrivial knots. The knots J and
K are called factor knots.
Def. A composition J#K of J and K is formed as follows:
remove a small arc from each knot projection
connect the endpoints of the removed arc of J
with the removed arc of K so that the added arcs
do not overlap and do not intersect the
remaining part of J and K .
Mth 333 – Spring 2013
Sections 1.1-1.2
1/8
Questions:
Note: J# (unknot) = J.
Mth 333 – Spring 2013
Sections 1.1-1.2
2/8
Oriented knots
Note: A knot can be oriented by consistently
orienting the segments making it up (or equivalently,
ordering the vertices).
Questions
1
How can we tell if a knot is prime or composite?
2
Is the unknot prime?
3
A knot is prime if it is not composite.
Can the same knots result in different
composites?
Note: Later, we will see the answer to (2) is no.
Note: Knot tables list only prime knots.
Mth 333 – Spring 2013
Sections 1.1-1.2
One way of forming a composite knot is to orient each
knot, and then connect the knots so that the
orientations match up.
3/8
Mth 333 – Spring 2013
Sections 1.1-1.2
4/8
Composition and Orientation
Composition and Orientation, II
Another way is to orient the knot and then connect
the knots so the orientations do not match up.
Mth 333 – Spring 2013
Sections 1.1-1.2
Connected sums using the first method yield the
same knot, no matter where the segments are
chosen.
5/8
Composition and Orientation III
Mth 333 – Spring 2013
Sections 1.1-1.2
Invertibility
Connected sums using the second method yield the
same knot, no matter where the segments are
chosen.
Def: A knot is invertible if it can be deformed back
onto itself so that a given orientation is taken to the
opposite orientation.
However: Different connected sums may result from
using method 1 in one composition and method 2 in
the other.
Note: If a knot K is invertible, then J#K is well
defined, no matter what orientations are used.
Mth 333 – Spring 2013
Sections 1.1-1.2
6/8
7/8
Mth 333 – Spring 2013
Sections 1.1-1.2
8/8