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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.

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Dror Bar-Natan 2003-10-15