Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory: | (28) |
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Assigned Thursday October 16; due Thursday October 23 in class.

This document in PDF: HW.pdf

**Required reading. ** Sections 1 and 2 of my paper
On the Vassiliev Knot
Invariants.

**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,
).
- Does 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
primitives.*

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Dror Bar-Natan 2003-10-15