Homework Assignment 6: Deframing

Assigned Thursday October 23; due Thursday October 30 in class.

This document in PDF: HW.pdf

Required reading. Sections 2 and 3 of my paper On the Vassiliev Knot Invariants.

Let be the multiplication operator by the chord diagram , and let be the adjoint of multiplication by on , where is the obvious dual of in . Let be defined by

The following assertions can be verified:

1. , where is the identity map and where for any two operators.

2. is a degree 0 operator; that is, for all .

3. satisfies Leibnitz' law: for any .

4. is an algebra morphism: and .

5. satisfies the co-Leibnitz law: (why does this deserve the name the co-Leibnitz law''?).

6. is a co-algebra morphism: (where is the co-unit of ) and .

7. and hence , where is the ideal generated by in the algebra .

8. If is defined by

then for all .

9. .

10. descends to a Hopf algebra morphism , and if is the obvious projection, then is the identity of . (Recall that .)

11. .

To be handed in. Verify assertions 4, 5, 7 and 11 above.

Recommended for extra practice. Verify all the other assertions above.

Idea for a good deed. Prepare a beautiful TEX writeup (including the motivation and all the details) of the solution of this assignment for publication on the web. For all I know this information in this form is not available elsewhere.

The generation of this document was assisted by LATEX2HTML.

Dror Bar-Natan 2003-10-22