AKT-14/Homework Assignment 1

DBN: Classes: 2013-14: AKT-14 / Navigation # Week of... Notes and Links 1 Jan 6 About This Class (PDF).  Monday: Course introduction, knots and Reidemeister moves, knot colourings. Tricolourability without Diagrams  Wednesday: The Gauss linking number combinatorially and as an integral.  Friday: The Schroedinger equation and path integrals. Friday Introduction (the quantum pendulum) 2 Jan 13 Homework Assignment 1.  Monday: The Kauffman bracket and the Jones polynomial.  Wednesday: Self-linking using swaddling.  Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres. 3 Jan 20 Homework Assignment 2.  Monday: The definition of finite-type and some examples.  Wednesday: The self-linking number and framings.  Friday: Integrating a polynomial times a Gaussian. Class Photo. 4 Jan 27 Homework Assignment 3.  Monday: Chord diagrams and weight systems.  Wednesday: Swaddling maps and framings, general configuration space integrals.  Friday: Some analysis of $d^{-1}$. 5 Feb 3 Homework Assignment 4.  Monday: 4T, the Fundamental Theorem and universal finite type invariants. The Fulton-MacPherson Compactification (PDF).  Wednesday: The Fulton-MacPherson Compactification, Part I.  Friday: More on pushforwards, $d^{-1}$, and $d^{\ast }$. 6 Feb 10 Homework Assignment 5.  Monday: The bracket-rise theorem and the invariance principle.  Wednesday: The Fulton-MacPherson Compactification, Part II.  Friday: Gauge fixing, the beginning of Feynman diagrams. R Feb 17 Reading Week. 7 Feb 24  Monday: A review of Lie algebras.  Wednesday: Graph cohomology and $\Omega _{dR}^{\ast }(\Gamma )$.  Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF). 8 Mar 3 Homework Assignment 6 (PDF)  Monday: Lie algebraic weight systems.  Wednesday: Graph cohomology and the construction of $Z_{0}$. Graph Cohomology and Configuration Space Integrals (PDF)  Friday: Gauge invariance, Chern-Simons, holonomies. Mar 9 is the last day to drop this class. 9 Mar 10 Homework Assignment 7 (PDF)  Monday: The $gl(N)$ weight system.  Wednesday: The universal property, hidden faces.  Friday: Insolubility of the quintic, naive expectations for CS perturbation theory. 10 Mar 17 Homework Assignment 8 (PDF)  Monday: $W_{\mathfrak {g}}\colon {\mathcal {A}}(\uparrow )\to {\mathcal {U}}({\mathfrak {g}})$ and PBW.  Wednesday: The anomaly.  Friday: Faddeev-Popov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF). 11 Mar 24 Homework Assignment 9 (PDF)  Monday: ${\mathcal {A}}$ is a bi-algebra.  Wednesday: Understanding and fixing the anomaly. Friday: class cancelled. 12 Mar 31 Monday, Wednesday: class cancelled.  Friday: A Monday class: back to expansions. E Apr 7  Monday: A Friday class on what we mostly didn't have time to do. Add your name / see who's in! Dror's Notebook

This assignment is due in class on Monday January 20. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment was written on the HW session of Friday January 10. See BBS/AKT14-140110-191441.jpg, BBS/AKT14-140110-192041.jpg, and BBS/AKT14-140110-192042.jpg.

Question 1.

A. Prove that the set of all 3-colourings of a knot diagram is a vector space over ${\mathbb {F} }_{3}$. Hence $\lambda (K)$ is always a power of 3.

B. Prove that $\lambda (K)$ is computable in polynomial time in the number of crossings of $K$.

Question 2. Using combinatorics alone prove that given a 2-component link diagram $D$, with components $\gamma _{1}$ and $\gamma _{2}$,

${\frac {1}{2}}\sum _{\scriptstyle {\text{xings }}x{\text{ between}} \atop \scriptstyle \gamma _{1}{\text{ and }}\gamma _{2}}(-1)^{x}$ $=\sum _{\scriptstyle {\text{xings }}x{\text{ with}} \atop \scriptstyle \gamma _{1}{\text{ over }}\gamma _{2}}(-1)^{x}$ $=\sum _{\scriptstyle {\text{xings }}x{\text{ with}} \atop \scriptstyle \gamma _{2}{\text{ over }}\gamma _{1}}(-1)^{x}$

Question 3. Use the techniques from Friday January 10, along with that Friday's handout (the quantum pendulum), to write $U_{t}=e^{-itH}$, with $H=-{\frac {1}{2}}\Delta +{\frac {1}{2}}x^{2}$, in explicit terms, for $0<t<{\frac {\pi }{2}}$. Note that $U_{t}$ interpolates between the identity and the Fourier transform.