# 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 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 is a type invariant of 2-component links then is a type invariant of knots.
2. Find a map (sorry for the operator overloading'') for which for all such and . (Verify that you proposed map respects the relation!)

2. If is a chord diagram, let be the number of chord crossings'' in (so for example, ).
1. Does satisfy the relation?
2. 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