Research-Related

Question 1

Whitney Disk

Answer 1

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

Question 2

Symmetric Product

Answer 2

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

Question 3

Morse function

Answer 3

f: Mn = R is a morse function if all the critical points are non-degenerate
(Hessian is non-singular).

Question 4

Index of critical point

Answer 4

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.

Question 5

Handle Cancellation

Answer 5

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.

Question 6

n dimensional k-handle

Answer 6

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

Question 7

Core/Cocore of a handle

Answer 7

For a handle D^k * D^(n-k) the core is D^k * 0 and the cocore is 0*D^(n-k)

Question 8

Attaching Sphere

Answer 8

For a handle D^k * D^(n-k) the attaching sphere is d(D^k) * 0

Question 9

Belt Sphere

Answer 9

For a handle D^k * D^(n-k) the belt sphere 0*d(D^(n-k))

Question 10

Handle slide

Answer 10

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.

Question 11

Geometrization Conjecture

Answer 11

Every closed 3-mfld can be canonically decomposed into pieces that have one of the 8 types (Spherical, Euclidean, Hyperbolic

Question 12

Correction Term

Answer 12

d(Y) = minimum dimension of any homogeneous element of HF^+ from HF^infinity.
This is an invariant for homology three-spheres

Question 13

Spin^c-structures

Answer 13

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.

Question 14

Holomorphic Disk

Answer 14

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.

Question 15

Seifert Surface

Answer 15

A surface (2-mfld) whose boundary is the given knot or link.

Question 16

Genus of a knot

Answer 16

The minimal genus of all Seifert Surfaces of a knot

Question 17

4-Ball Genus

Answer 17

The minimal genus of all 2-mflds bounded the knot smoothly embedded into the four ball

Question 18

Fibered Knot

Answer 18

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.

Question 19

If S3_{1/q}(K) is Seifert Fibered…

Answer 19

All non-zero torsion coefficients have the same sign

Question 20

Knot Signature

Answer 20

The signature of the Seifert matrix, which is computed from the Seifert surface.

Question 21

Casson’s Invariant

Answer 21

Surjective map lambda from oriented integral homology 3-spheres to Z satisfying:

• lambda(S3) = 0
• Unique up to multiplication by constant

Question 22

Alexander Polynomial

From HFK

Answer 22

Sum Chi(HFK(K,i)* T^i

Question 23

Knot genus

From HFK

Answer 23

max{i, hat[HFK(K,i)] neq 0}

Question 24

If degree of alexander polynomial is less than the genus…

Answer 24

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

Question 25

S3_{p/q}(K) has ____ many spin^c structures

Answer 25

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

Question 26

Correction Term d(Y)

Answer 26

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.

Question 27

For large p Sp3(K) is given by

Answer 27

Hk(A_s^0)

Again splitting over spin^c structures

Question 28

Descending Manifold (with morse function)

Answer 28

Set of all points with flows “away” from a critical point

beta curves are intersections of Sigma with descending manifold

Question 29

Ascending Manifold (with morse function)

Answer 29

Set of all points with flows “toward” a critical point

alpha curves are intersections of Sigma with ascending manifold

Question 30

Tau invariant

Answer 30

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

Question 31

Chern Class

Answer 31

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.