Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory:

(28)

Next: Class Notes for Thursday October 16, 2003 Previous: Class Notes for Tuesday October 14, 2003

Homework Assignment 5

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.

1. Let $\Delta$ 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.
   1. Show that if $V$ is a type $m$ invariant of 2-component links then $V\circ\Delta$ is a type $m$ invariant of knots.
   2. Find a map $\Delta:{\mathcal A}(\bigcirc)\to{\mathcal A}(\bigcirc\bigcirc)$ (sorry for the ``operator overloading'') for which $W_{V\circ\Delta}=W_V\circ\Delta$ for all such $m$ and $V$. (Verify that you proposed map respects the $4T$ relation!)

2. If $D$ is a chord diagram, let $X(D)$ be the number of ``chord crossings'' in $D$ (so for example, $X(\otimes)=1$).
   1. Does $X:{\mathcal D}\to{\mathbb{Z}}$ satisfy the $4T$ relation?
   2. Let $m$ by a natural number. Can you find a type $m$ knot invariant $V$ for which $W_V=X$?

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.

The generation of this document was assisted by L^ATEX2HTML.