Research-Related Flashcards
Whitney Disk
Given a pair of intersection points x, y in Talpha, Tbeta, a Whitney Disk between x and y is a continuous map u: D to Symg such that u(-i)=x and u(i)=y.
D is the unit disk in C
Symmetric Product
For a genus g surface, symmetric product
Sigma_g * … * Sigma_g mod Sg, the symmetric group on g.
It consists of unordered g tuples of points in Sigma_g
It is a smooth manifold
Morse function
f: Mn = R is a morse function if all the critical points are non-degenerate
(Hessian is non-singular).
Index of critical point
The index of a critical point is the number of negative entries in the diagonal of the hessian once diagonalized.
Thought of as the number of directions of flow from a critical point.
Handle Cancellation
A (k-1)handle h_k-1 and a k-handle hk can be cancelled provided that the attaching sphere of hk intersects the belt sphere of h(k-1) transversally in a single point.
n dimensional k-handle
A copy of D^k * D^(n-k) attached to the boundary of an n-mfld X along d(D^k)*D^(n-k)\
0-handle attached by disjoint union
Core/Cocore of a handle
For a handle D^k * D^(n-k) the core is D^k * 0 and the cocore is 0*D^(n-k)
Attaching Sphere
For a handle D^k * D^(n-k) the attaching sphere is d(D^k) * 0
Belt Sphere
For a handle D^k * D^(n-k) the belt sphere 0*d(D^(n-k))
Handle slide
Given two handles h1 and h2 attached to d(X) a handle slide of h1 over h2 is where we isotope attaching sphere of h1 in d(X cup h2) by pushing it through belt sphere of h2.
Geometrization Conjecture
Every closed 3-mfld can be canonically decomposed into pieces that have one of the 8 types (Spherical, Euclidean, Hyperbolic
Correction Term
d(Y) = minimum dimension of any homogeneous element of HF^+ from HF^infinity.
This is an invariant for homology three-spheres
Spin^c-structures
Equivalence classes of intersection points. They are in affine correspondence with H^2 = H_1. There are p-many Spin^c structures for p/1 surgery on L.
Holomorphic Disk
A complex differentiable function. M(phi) = equivalence classes of holomorphic representatives of phi in pi_2(x,y). The massive grading is expected dimension of M(phi).
We only look at disks that have holomorphic representatives.
Seifert Surface
A surface (2-mfld) whose boundary is the given knot or link.
Genus of a knot
The minimal genus of all Seifert Surfaces of a knot
4-Ball Genus
The minimal genus of all 2-mflds bounded the knot smoothly embedded into the four ball
Fibered Knot
A 1-parameter family of Seifert surfaces for K such that the intersection of any two different SS’s in this family have intersection exactly the knot.
Fibered knots always have monic alexander polynomial.
If S3_{1/q}(K) is Seifert Fibered…
All non-zero torsion coefficients have the same sign
Knot Signature
The signature of the Seifert matrix, which is computed from the Seifert surface.
Casson’s Invariant
Surjective map lambda from oriented integral homology 3-spheres to Z satisfying:
- lambda(S3) = 0
- Unique up to multiplication by constant
Alexander Polynomial
From HFK
Sum Chi(HFK(K,i)* T^i
Knot genus
From HFK
max{i, hat[HFK(K,i)] neq 0}
If degree of alexander polynomial is less than the genus…
1/q surgery on the knot is NEVER a positively oriented SFS.
If in addition g(K) > 1 then no 1/q surgery along K is Seifert fibered
S3_{p/q}(K) has ____ many spin^c structures
It has p many spin^c structures.
So to get SFHS we must have p=1
If p=0 there are infinitely many spin^c structures
Correction Term d(Y)
minimum dimension of any homogenous element of HF+(Y) from HFinfinity(Y)
- d(S3) = 0
- If Y is an integral homology 3-sphere which bounds a smooth negative definite four-manifold then d(Y) >= 0
- d(S^3_1(K)) is always even
- Correction term gives us grading of the bottom of the tower.
For large p Sp3(K) is given by
Hk(A_s^0)
Again splitting over spin^c structures
Descending Manifold (with morse function)
Set of all points with flows “away” from a critical point
beta curves are intersections of Sigma with descending manifold
Ascending Manifold (with morse function)
Set of all points with flows “toward” a critical point
alpha curves are intersections of Sigma with ascending manifold
Tau invariant
minimum m in Z such that the map
i: Hk(F(K,m)) to Hk (hat(CF)(S^3)) = Z
is non trivial.
F is the set of all points with filtration level less than m
Chern Class
Think of as 2nd cohomology classes on the four-mfld First chern classes of spin^c structures are of form x+2h where h is a fixed coho class and x only depends on underlying 4-mfld.
