Math 104 Winter 2009

Topics covered and homework assignments.

Math 104 Main page

Date	Chapter and material description	Homework assignment
Monday January 5	Definitions and examples of manifolds, introducing the concept of a knot	Read: class notes
Wednesday January 7	Knots, PL Knots, Smooth knots, Isotopy, Smooth Isotopy, Ambient isotopy	Read class notes and pages 5-9 of the textbook
Friday January 9	Connected sum of knots, Reidemeister moves, transversality and the submanifold parameterizing the intersection of two transverse knots	Read class notes and pages 10-14 of the textbook. Exercise 1.2 part a only and Exercise 1.4 on page 14 due Wednesday January 14 in written form.
Monday January 12	Computation of singularity codimension, Reidemeister moves	Read class notes. Compute the codimension of the singularity of the knot projection at which three branches pass though one point on the plane and two of the three branches are tangent in written form due Thursday January 22
Wednesday January 14	Smooth and immersed curves, rotation number, immersed curves without direct tangencies and knots in a solid torus	There are 4 types of kinks on oriented knot diagrams. They differ by their input into the writhe and into rotation numbers. Show that two kinks whose inputs into both rotation and writhe are opposite can be cancelled by a sequence of second and third Reidemeister moves, in written form due Thursday January 22
Friday January 16	Covering spaces and their examples. Every knot type in R^3 can be realized by the lift of an immersed curve of rotation number zero to the total space of the universal covering: R^3=R^1 ×D^2→ST R^2	Present an example of an immersed curve of rotation number zero that lifts to the trivial knot and another one that lifts to a trefoil in written form due Thursday January 22
Monday January 19 Martin Luther King Jr. Day. No class Note that Tuesday January 20 is the final day for electing the Non-recording option
Wednesday January 21	Fundamental group and higher order homotopy groups	Use the cover S^3→RP^3and the fact that π3 (S^3, x0)=Z to compute π3 (RP^3, y0) in written form due Wednesday January 28
Thursday January 22 x-hour instead of the class on Monday January 19	More on fibrations and their exact homotopy sequences, h-principle for immersions	Use the universal covering of the wedge of two circles to compute all the homotopy groups of it of order i≥2 in written form due Wednesday January 28
Friday January 23	Smale’s computation of the homotopy groups of the space of based immersions of a circle into a surface	Read the notes carefully. Next time we will apply the results of this lecture and the techniques of computing codimensions of the subspaces of singular curves to define Arnold invariants
Monday January 26	Construction of Arnold’s J^+invariant of immersed curves on the plane	Read the class notes carefully.
Wednesday January 28	Construction of Arnold’s J^+invariant, wave fronts and their relation to knot theory.	Prove that for every integers N, R there exists a curve of rotation number R whose J^+invariant is larger than N. Note that you may have to consider separately the cases R=0, +1, -1, in written form due Wednesday February 4
Friday January 30	Viro’s integral formula for the J^+invariant	For a natural number n, let Cn be an oriented planar immersed curve that starts as an outwards going counterclockwise oriented spiral, does n turns and closes up by returning to the spiral center along a straight line segment. Use Viro integral formula to compute J^+ (Cn), in written form due Wednesday February 4
Monday February 2 The Middle of the term presentation and discussion should be done in the period Monday February 2- Sunday February 8	Viro’s integral formula for J^+ invariant, contact structures	Read the lecture notes carefully.
Wednesday February 4	Contact Structures and Legendrian knots	Draw a front projection of a Legendrian knot that is topologically a trefoil knot in written form due Wednesday February 11
Friday February 6	Contact structures on the ST^*M, Lorentz manifolds, time orientation	Recall that the selflinking number slk(K) of a framed oriented knot K is defined as the linking number of the two component oriented link formed by the knot K and the other knot obtained by shifting K slightly along the framing vectors. Let K1 and K2 be framed knots in R^3that are smoothly isotopic as unframed knots. Prove that K1 and K2 are isotopic as framed knots if and only if slk(K1)= slk(K2) in written form due Wednesday February 11
Monday February 9	Globally hyperbolic spacetimes, Geroch Theorem, Bernal Sanchez Theorem. Cauchy surfaces their spherical cotangent bundles and the space of all null geodesics. Low conjecture and the Legendrian Low conjecture due to Natario and Tod	Let M be the 2-dimensional Minkowski spacetime, i.e. it is R^2 equpped with the Lorentz metric dx^2-dt^2. Exercise 1: Give three examples of different Cauchy surfaces in M, two of them spacelike and smooth and the third non smooth. Note your Cauchy surfaces are one dimensional. Exercise 2: Give three examples of curves in M that are not Cauchy surfaces for various reasons. in written form due Wednesday February 18
Wednesday February 11	Jones polynomial, Kauffman bracket	Compute the Jones polynomial of the knot 41 shown in Figure 3.13 on page 33 using the formula X(L)=(-A)^-3w(L)<L> and the substitution A=q^-1/4in written form due Wednesday February 18
Thursday February 12 x-hour instead of the class on Friday February 13	State sums and the skein relation for the Jones polynomial	Compute the Jones polynomial of the knot 41 shown in Figure 3.13 using the skein relations on pages 29-30. Note that if at a certain moment you get a trivial link with the nontrivial diagram you can apply axioms 2’ and 3’. In written form due Wednesday February 18
Friday Feburary 13 Winter Carnival! No class Final day for dropping a fourth course without a grade notation of "W"
Monday February 16	Jones polynomial for the mirror image of the link, and for connected sum of links. Vassiliev invaraints.	Carefully read the class notes.
Wednesday February 18	Vassiliev-Goussarov invariants, Kontsevich Theorem: pages 36-42	Find all the homotopy classes of singular knots with 3 double points in terms of the corresponding chord diagrams. Use the relations we today discussed in class to get the relations on the values of the 3^rd order derivatives of real valued Vassiliev invariants of order ≤3 on these diagrams. See pages 42-43. In written form due Wednesday February 25
Friday February 20	More on Vassiliev-Goussarov invariants, chord diagrams prepresentation of homotopy classes of singular knots. Gauss diagram of a knot	Draw Gauss diagrams of the knots 41 and 51 on page 33 of the textbook. In written form due Wednesday February 25
Monday February 23 Note that Tuesday February 24 is the last day to withdraw from the course	Virtual knots, Jones polynomial and Kauffman bracket of a virtual knot. Polyak-Viro formulas for Vassiliev invariants of a knot K in terms of a Gauss diagram of K	Use Polyak-Viro formulas to compute the order two Vassiliev invariant of the knots knots 41 and 51 on page 33 of the textbook. Compute the Jones polynomial of the virtual knot on the last page of the today class handout. In written form due Wednesday March 4
Wednesday February 25	Polyak-Viro formula for the order 2 Vassiliev invariant. Braid groups	Problem 5.1 on page 51 In written form due Wednesday March 4
Thursday February 26 x-hour	Alexander and Markov Theorems	Read the lecture notes carefully
Friday February 27	Pure braids, Thickened braids, automorphisms of a disk with holes that are identity on the boundary considered modulo isotopy	Read the lecture notes carefully
Monday March 2 Note that Tuesday March 3 is the final day to alter grade limit filed under the Non-Recording Option	Heegaard decomposition for closed 3-manifolds and for 3-manifolds with boundary	Read the lecture notes carefully
Wednesday March 4	Lens Spaces	Read the lecture notes carefully
Friday March 6	Lens spaces and the Theorem about being able to obtain every closed 3-manifold by cutting out the solid tori for a 3-sphere and gluing them back	Problem 12.4 on page 87 Hint: you might want to use the statement of Lemma 12.5 that we prove on Monday In Written form is due on Monday March 9
Monday March 9 The end of the term oral presentation and general discussion should be done during the period of Monday March 9 – Saturday March 15	Lemma 12.5 and the Rokhlin’s Theorem