Homework Assignment 5
Assigned Thursday October 16; due Thursday October 23 in class.
Required reading. Sections 1 and 2 of my paper
On the Vassiliev Knot
To be handed in.
- Let be the ``doubling'' (also called ``cabling'') operation
on knots, which takes a framed knot and replaces it by a 2-component
link by ``replacing every line by a double line'' in an obvious manner.
- Show that if is a type invariant of 2-component links then
is a type invariant of knots.
- Find a map
(sorry for the ``operator overloading'') for which
for all such and . (Verify that
you proposed map respects the relation!)
- If is a chord diagram, let be the number of ``chord
crossings'' in (so for example,
satisfy the relation?
- Let by a natural number. Can you find a type knot invariant
for which ?
Idea for a good deed. Tell us about the
Milnor-Moore theorem: A connected commutative and co-commutative graded
Hopf algebra over a field of characteristic 0 which is of finite
type, is the symmetric algebra over the vector space of its
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