Definitions Flashcards
(111 cards)
1
Q
Unit Circle
A
S^1 = {z ∈ C : |z| = 1}
2
Q
f: S^1 → R^3 is smooth
A
Let f: S^1 → R^3 be a map.
We call f smooth if all its derivatives exist. More precisely, all derivatives of the map g: R → R^3, g(t) = f(exp(2πit)) exist.
3
Q
f: S^1 → R^3 is immersive
A
f is immersive if it’s smooth and g’(t) =/= 0 for all t ∈ R.
Where g: R → R^3, g(t) = f(exp(2πit))
4
Q
f: S^1 → R^3 is an embedding.
A
f is an embedding if it is smooth, injective and immersive. (It’s image is a knot).
5
Q
Knot
A
The image of an embedding f: S^1 → R^3.
6
Q
A link
A
A link is a union of finitely many disjoint knots.
7
Q
A component of a link
A
One of the knots in the finite union.
8
Q
A Projection
A
p_12:R^3 → R^2, (x,y,z)→(x,y)
9
Q
Curves in general position (Shadows)
A
Let d: S^1 × {1, . . . , k} → R2.
d and its image are in general position (shadows) if:
(1) d is immersive
(2) d^-1 (a) ≤ 2 for all a ∈ R^2
(3) Use d′(exp(2πit), i) as short-hand for (d/dt) d(exp(2πit), i) where 1 ≤ i ≤ k.
Let x, y ∈ S^1 × {1, . . . , k} be distinct such that d(x) = d(y). Then d′(x), d′(y) are linearly independent over R.
10
Q
Topological knot
A
Let f: S^1 → R^3 be a continuous injective map, then it’s image is a topological map.
11
Q
Crossing / Ordinary double point of d
A
Let d: S^1 × {1, . . . , k} → R^2 b e in general position.
If d(x)=d(y) and x =/= y then d(x) is a crossing of d.
12
Q
A knot diagram.
A
A knot diagram is a pair (d, A), d: S^1 × {1, . . . , k} → R^2 in general position, with under/over data A.
A chooses one of {x,y} if d(x)=d(y), x=/=y.
[ If f:S^1→ R^3 is an embedding, then p_12 ◦ f in general position gives rise to a knot diagram (d, A) where d = p_12 ◦ f and the over/under data A picks x or y according to the greater among p3 ◦ f(x) and p3 ◦ f(y). ]
13
Q
Orientation Preserving
A
r: R^3 → R^3 is orientation preserving if if the determinant of the jacobian (∂r_j/∂x_i) is positive everywhere.
14
Q
A diffeomorphism
A
A diffeomorphism is a smooth bijection whose inverse is smooth.
15
Q
Isotopic (horrible definition)
A
Let f , g: S^1 → R3 be embeddings. The following are equivalent:
1) There exists a continuous map h: [0, 1] × S^1 → R^3
such that the map h_t: S^1 → R^3, h_t(x) = h(t, x) is an embedding for all t ∈ [0, 1], and (f , g) = (h_0, h_1).
2) There exists an orientation preserving diffeomorphism r: R^3 → R^3 that by restriction defines an orientation preserving bijection f(S^1) → g(S^1).
If these equivalent conditions are satisfied then we say that the oriented knots f(S^1) and g(S^1) (with the standard orientation) are isotopic. We also say that f and g are isotopic.
16
Q
Combinatorially Equal
A
Two link diagrams (d,A) (e,B) are combinatorially equal if there exists an orientation preserving diffeomorphism r: R^3 → R^3 such that e = r ◦ d and carrying A over to B.
17
Q
Reidermeister Moves
A
(R1) (R2) (R3)
18
Q
Isotopic
A
Two link diagrams are isotopic if one can be obtained from the other by a sequence of reidermeister moves.
19
Q
Types of crossing
A
Positive (Top arrow right to left) sign=1
Negative (Top arrow left to right) sign = -1
20
Q
Linking Number of components
A
The linking number L_k(C_1,C_2) for components C_1 and C_2 of a link is half the sum of the signs of the crossings where C_1 meets C_2.
21
Q
Link invariant.
A
A link invariant is a map f: {isotopy classes of links} → X for any set X.
22
Q
Writhe
A
The Writhe w(D) of an oriented knot diagram is the sum of the signs of the crossings.
23
Q
Colouring
A
Let (A,+) be an abelian group.
Let D be an unoriented link diagram.
A colouring of D with values in A is a non-constant map f:arcs(D)→A such that at each crossing:
a - 2b + c = 0,
where b is the value of the overpassing arc and a, c, the other two arcs.
24
Q
n-colouring
A
An n-colouring is a colouring with values in Z_n
25
n-colourable
A link diagram is n-colourable with colouring number n if there exists a (non-constant) n-colouring.
26
Arc
Let (d,A) be a link diagram.
An arc is d(I) where I is a maximal open connected subset of S^1 × {1, . . . , k} such that d(I) does not contain under passings.
27
(1) Split
| (2) Splittable
A link diagram is split if it meets both half planes x_1 0 but not x_1 = 0 in between.
A link diagram is splittable if it is isotopic to a split one
28
Alternating link diagram
A link is alternating if it admits an alternating diagram.
29
Types of crossing in a chess-boarded link diagram
Type 1: Vertical line is over (top right/bottom left coloured)
Type 2: Horizontal line is over (top right/bottom left coloured)
30
Abelian group assiciated with A
Let A ∈ Z^(m,n). The abelian group associated with A is:
G(A) := Z^(1,n)/Z^(1,m) A ∼= Z(n,1)/A^T Z^(m,1)
Any finitely generated abelian group is of this form up to isomorphism.
31
Generators and Relations of G(A) (the abelian group associated with A).
Generators {x_j s.t 1≤ j ≤ n}
| Relations ∑_j a_(i j) x_j = 0 for all i.
32
A unimodular matrix
A unimodular matrix is a matrix P ∈ Z^(n,n) such that det(P) ∈ {−1, 1} or equivalently P^−1 ∈ Z^(n,n).
33
Z-equivalent
Let A, B ∈ Z^(m,n).
We say that A, B are Z-equivalent if there are unimodular P, Q such that B = PAQ.
Equivalently, B can be obtained from A by a sequence of
unimodular elementary row and column operations.
34
Smith Normal Form
Every equivalence class contains a unique Smith normal form, that is, a matrix B = (bi j) such that
◦ b_(i j) = 0 unless i = j.
◦ b_(ii) ≥ 0 for all i.
◦ b_(ii) | b_(i+1,i+1) for all i.
35
Colouring group.
| Reduced colouring matrix
Let D be a diagram for a link L. Let A be the colouring matrix of D.
Let B be a reduced colouring matrix, a matrix obtained from A by removing one row and one column.
The colouring group of L is
Col(L) := G(B) = Z^n / Z^n B.
The determinant of L is det(L) := |det(B)|.
36
Alexander Polynomial
Let H = ∈ Z [t, t^-1] (group of invertible elements in Z [t, t^-1] ).
The Alexander polynomial is an invariant of oriented links whose values are in Z [t, t^-1] / ∼ where where f ∼ g means fH = gH .
Denoted: ∆_L
37
Alexander Matrix
The alexander matrix has rows indexed by crossings and columns indexed by arcs. Entries are a + (t - 1)b - tc (where b is the overpassing).
38
Reverse rL
Mirror Image mL
Reverse rL is L with opposite orientation.
Mirror image mL is L with the reflection (x,y)→(x,-y).
39
(topological) n-manifold.
Let n ≥ Z_(≥0). A (topological) n-manifold is a Hausdorf topological space M such that for all x ∈ M there exists an open set U ⊂ M containing x such that U is homeomorphic to R^n or R^(n−1) × R_(≥0).
40
Interior Point of an n-maifold
| Boundary Point of an n-manifold
Let M be an n-manifold and x ∈ M. If there exists an open set U ⊂ M containing x such that U is homeomorphic to R^n, then x is called an interior point of M.
Otherwise x is a boundary point. The set of boundary points is called the boundary of M and written ∂M. We write int(M) = M \ ∂M.
41
Closed n-manifold
A n-manifold is said to be closed if it is compact and its boundary is empty.
(This should not be confused with closed subsets of topological spaces.)
42
An orientation on a surface
An orientation of a surface (=2-manifold) S is a continuous map S × ∆ → S whose restriction to {s} × ∆ is an embedding and taking (s, 1) to s for all s ∈ S.
(In other words, if you pull ∆ around the surface and you come back to a place you've been before then ∆ still has the same orientation.)
43
K
K={Isotopy classes of knots}
44
Knot sum
K × K → K written (K, L) → K + L
| defined by picture..
45
Factorising Sphere
∂T_1 = ∂T_2 ⊂ T_1 ∪_f T_2 a factorising
| sphere for the factorisation K_1 + K_2.
46
Monoid
Group without inverses
47
A prime knot
A prime knot is an oriented knot not of the form K+L unless K or L is the unknot.
48
S
S is the set of homeomorphism classes of non empty compact connected surfaces.
49
Connex Sum
S × S → S: (S, T) → S + T
50
Torus
Let T ∈ S be the class of the torus S^1 × S^1.
51
Disk
Let D ∈ S be the class of the disk {z ∈ C : |z| ≤ 1}.
52
Projective plane or cross-cap.
Let P ∈ S be the class of the projective plane or cross-cap P = R P^2.
53
S^+
S^+ is the subset of S consisting of the (homeomorphism classes of) orientable surfaces.
S^+ is a submonoid of S.
54
Presentations of monoids
S = ,
S^+ =
55
Genus g(S) of orientable surface S
Let S be an orientable surface of class gT + nD. Then g is called the genus g(S) of S.
56
Number of boundary points of gT+nD+mP
n
57
Spanning surface
Let L ⊂ R^3 be an oriented link. A spanning surface for L is a compact oriented surface S in R^3 whose (oriented) boundary is L.
58
Isotopy of spanning surfaces
Let L be an oriented link and S a compact oriented surface.
Let h: [0, 1] × S → R^3 be a continuous map such that, on
writing h_t (x) = h (t, x), the map h_t: S → R^3 is injective and its image spans L.
Then we say that the spanning surfaces h_0 (S) and h_1 (S) are isotopic (relative L).
59
Projection Surface
D diagram of oriented link L. The constructed spanning surface for L is called the projection surface.
60
Seifert Circle
Let D_s be the picture obtained from D by resolving or smoothing every crossing.
A circle in D_s is called a Seifert circle.
61
The depth of a Seifert circle
The depth D(c) of a seifert circle C is the number of Seifert circles it is properly enclosed by.
62
Genus of an oriented link.
Let L be an oriented link.
The genus, g(L) of L is the least genus of a connected spanning surface .
[ Genus = (#Seifert circles - #crossings - 1) / 2 ]
63
Adding/Removing a tube.
Let F be a spanning surface for an oriented link L.
Let D be a disk embedded in R^3 \ L such that D ∩ F = ∂D.
Cut F along D and attach copies of D to the boundary to obtain another spanning surface F′.
We say that F′ is obtained from F by removing a tube and F from F′ by adding a tube.
64
S-equivalent
Two spanning surfaces for an oriented link L are S-equivalent if one can be obtained from the other by repeatedly adding or removing a tube.
65
Band Crossing
| Band Flip
See pictures
66
Good orientation
Projection Surface
See picture
67
First homology group
Let F be a oriented surface. The first homology group H_1 (F) is the Z-module (equivalently, abelian group) C_1/D_1 where C_1 is the free Z-module on the set of oriented closed 1-manifolds in F and D_1 is the least submodule of C_1 such that:
◦ The submodule D_1 contains the boundary of any compact subsurface of F.
◦ We have (A + B) − C ∈ D_1 whenever A, B are disjoint oriented closed 1-manifolds in F and C is their union.
[A closed 1-manifold is the same as a topological space homeomorphic to S^1 × {1, . . . , k} for some natural number k]
68
Linking Number of disjoint oriented links
Their linking number is defined to be
L_k (L_1, L_2) = ∑ L_k (K_1, K_2)
where the sum is over all pairs (K_1, K_2) where K_i
is a component of L_i.
69
Seifert Matrix
Let F be a connected spanning surface of an oriented link L.
Consider a Z-basis A = (a_1, . . . , a_r) of H_1 (F).
Moving F slightly away from itself in positive direction gives a surface F^+ disjoint from F and a homeomorphism φ: F → F^+.
This induces an isomorphism
φ_∗: H_1 (F) → H_1(F^+).
Put a^*_i = φ_∗(a_i).
Recall the linking numbers of definition 19.
The Seifert matrix is:
M(F, A) = ( l_k(a_i, a^+_j) ) _ (i, j)
70
Seifert Polynomial
Let M be a Seifert matrix of a link L.
The Seifert polynomial of L is
S(L) = S_L (x) := det (xM − x^−1M^T) ∈ Z[x, x^−1].
71
T-equivalence
We define ∼_T (T-equivalence) to be the least equivalence relation on the set of square matrices over Z (of any size) such that:
(a) We have M ∼_T P^T M P whenever P is unimodular.
(b) We have
M ∼_T M ∗ 0
∗ ∗ a
0 b 0
whenever M is an n × n a square matrix over Z, the matrix on the right is (n + 2) × (n + 2), and a, b ∈ Z, a^2 + b^2 = 1, and a star ∗ means any contents
allowed.
72
µ(L)
µ(L) = #components of link L
73
Breadth
Let f = ∑_i=k^l a_i t^i with a_i ∈ Z for all i and a_k, a_ℓ =/= 0.
Then ℓ − k is the breadth of f .
Also, the breadth of 0 is 0.
74
Skein triple of diagrams/links
A skein triple of diagrams (D_−, D_0, D_+) is a triple of oriented link diagrams differing only in a disk where they look as follows:
D_- : *left over right*
D_0 : *straight down*
D_+ : *Right over left*
Replace diagram by its link for skien triple of links.
75
Configuration Space
We define configuration space C_n = {X ⊂ C | #X = n}.
We equip it with the base-point ∗_n := {1, . . . , n} ∈ C_n.
76
Geometric n-braid
Let γ: [0, 1] → C_n be continuous and such that γ(0) = γ(1) = ∗_n.
With γ we associate the geometric n-braid
b(γ) := {Union of γ(t) x{t}, 0≤t≤1}
= { (x,t) | t ∈ [0, 1], x ∈ γ(t) } ⊂ C × R = R^3.
77
Braid Diagram
Projection on coords 1 and 3 of b(γ) along with over/under data.
78
Homotopic and isotopic
Let h: [0, 1]^2 → C^n be continuous and write h_a (t) = h (a, t).
Assume that
h_a (0) = h_a (1) = ∗n for all a.
We say that h_0 and h_1 are homotopic and that
the geometric braids b(h_0) and b(h_1) are isotopic.
79
B_n
B_n = { isotopy classes of n-braids}
80
Multiplication of Braids
Done by stacking. left right = top bottom
81
σ_i = σ^(n)_i ∈ Bn
Let 1 ≤ i ≤ n − 1.
We define σ_i = σ^(n)_i ∈ Bn to be the braid with
the following diagram.
*i and i+1 cross (over right to left) *
82
Bridge of a link diagram
A bridge of a link diagram is an arc containing at least one over passing.
83
Bridge number
The bridge number b(L) of a link L is the least number of bridges in a diagram for it.
84
Tunnel
If D is a link diagram, let cD be the same diagram with all crossings changed. A tunnel of D is by definition a bridge of cD.
85
Circle bridge/tunnel
A circle bridge is a bridge homeomorphic to a circle. Likewise for circle tunnels.
86
Closure of a braid
The closure is the map cl: B_n → {isotopy classes of oriented links} defined by the following picture:
87
R∞
See picture
88
With a link diagram D in S^2 we associate a link in R^3
| (up to isotopy) as follows:
First remove any point ∞ ∈ S^2 \D.
| This gives us a link diagram in a copy of R^2. Then make this into a link the usual way.
89
Conflicting and Compatible Seifert circles.
Let D be an oriented link diagram in S^2.
Let X, Y be two distinct Seifert circles of D.
Then X, Y are disjoint and S^2\ (X ∪ Y) has three components.
Precisely one of them is an annulus A (that is, A ∼= S^1 × (0, 1)).
If X, Y define the same orientation on A then X, Y are said to conflict. Otherwise they are compatible.
90
χ(D)
χ(D)= # unordered pairs of distinct conflicting Seifert circles
91
Reduction Move
Let D, D′ be oriented link diagrams in S^2.
We say that D → D′ is a reduction move if they are the same outside a small disk where they differ as in
the pictures below, provided the strings a, b belong to different Seifert circles.
92
B
B = ∪_(n≥0) {n} × B_n = { (n, x) | n ≥ 0, x ∈ B_n}
93
Markov Moves
Let ∼ be the least equivalence relation on B such that:
```
◦ Markov move 1. #
We have (n, a) ∼ (n, b) for all n ≥ 0 and conjugate a, b ∈ B_n.
(that is, b = x a x^−1 for some x ∈ B_n).
```
```
◦ Markov move 2.
We have (n, c) ∼ (n + 1, cσ_n) ∼ (n + 1, cσ^−1_n) for all n ≥ 0 and c ∈ B_n.
```
94
Conway Polynomial
The conway Polynomial is a unique polynomial CL(z) ∈ Z[z] such that
S_L (x) = C_L (x^(−1) − x).
95
A unique invariant of nonempty unoriented link diagrams D → [D] with values in R satisfies:
Bracket Polynomial
Let a, b, c be elements of a commutative ring R with 1.
Then there is a unique invariant of nonempty unoriented link diagrams D → [D] with values in R satisfying
1) [U_1] = 1 (unknot)
2) [ cross left over right ] = a [ = ] + b [ )( ]
3) [O D] = c [D]
The above identities are skein relations.
Let a = t, b = t^(-1) and c = -(t^(-2) + t^2)
Then [D] is the bracket polynomial.
If D is an oriented link diagram and D_o the underlying unoriented link diagram then [D] := [D_o]
96
State
Let D be an unoriented link diagram. Let C_D denote the set of crossings of D.
A state is a function f: C_D → {1, 2}.
For every state f we shall define a diagram without crossings D(f).
To obtain D(f), every crossing X ∈ C_D is replaced by Yf(X) as in the following picture.
X = *crossing left over right*
Y1 = *horizontal curves*
Y2 = )(
97
State Sum
```
For a state f: C_D → {1, 2} write
T(D, f) = a^k b^ℓ c^m
where k = # f^(−1) (1) , ℓ = # f^(−1) (2) and m is the number of components of D(f).
The state sum formula is
[D] = Σ_f T(D, f)
where the sum is over the states f.
```
98
X(L)
For an oriented link L we define
X(L) = X_L(t) = (−t)^(−3 w(D) ) [D]
for any diagram D of L.
99
Jones Polynomial
For an oriented link L we define the Jones polynomial
V_L (q) := X_L ( q^(−1/4) ).
From now on we write q = t^−4.
So (a, b, c) = ( q^(−1/4), q^(1/4), −(q^(1/2) + q^(−1/2) ).
Note that the Jones polynomial of the unknot is 1.
100
The Homfly Polynomial
There are unique polynomials
P(L) = P_L(q, z) ∈ Z[q, q^(−1), z],
one for each isotopy class of nonempty oriented links L, such that:
◦ For the unknot U_1 we have P(U_1) = 1.
◦ For any skein triple of links (L_−, L_0, L_+) we have
q^(−1) P(L_+) − q P(L_−) = z P(L_0).
It is known as the homfly polynomial.
101
Braid index
Let L be an oriented link. The braid index i(L) of L is the least natural number n such that L is the closure of some n-braid.
102
(a) Descending oriented knot diagram
| (b) Descending oriented link
(a) An oriented knot diagram is said to be descending if one can draw the associated shadow in the positive direction without lifting the pen such that every overpassing is drawn before the underpassing is.
The point where the drawing starts and finishes is called the base point.
(b) An diagram D of an oriented link L is said to be descending if each of its components is descending in the foregoing sense, and there is a total ordering
on the set of components K_D = K_L of L such that if P, Q ∈ K_D are distinct and P
103
s(D)
The number of Seifert circles of a link diagram D
104
```
Let D be a descending (oriented) link diagram.
Then s(D) + w(D) ≥ ......
```
s(D) + w(D) ≥ µ(D).
105
k = t-mindeg P and ℓ = t-maxdeg P.
Let R be a commutative ring and P ∈ R[t, t^(−1)] a nonzero Laurent polynomial over R in a variable t.
Say,
P = ∑^ℓ_(i=k) a_i t^i
with a_k, a_ℓ nonzero.
We write
k = t-mindeg P and ℓ = t-maxdeg P.
Also, t-mindeg(0) = ∞ and t-maxdeg(0) = −∞.
We put
t-maxdeg(P) − t-maxdeg(P) = t-breadth(P)
if P =/= 0 and t-breadth(0) = 0.
Likewise for any other variable.
106
The matrix S(i, n) ∈ GL(n, Z[x, x^(−1)] )
For 1 ≤ i ≤ n, let S(i, n) ∈ GL(n, Z[x, x^(−1)] ) be the matrix defined by the block form
S(i, n) =
I_i−1 · · ·
· 1 − x^2 x ·
· x · ·
· · · I_(n−1−i)
107
Burau Representation
There exists a unique representation (group homomorphism)
βn: Bn → GL(n, Z[x, x^(−1)] )
such that β_n (σ_i) = S(i, n) for all i.
It is known as the Burau representation.
108
Row Vector
We define the row vector v_n = (1, x, . . . , x^(n−1) ).
109
β_n (a)
Let a ∈ B_n.
We write β_n (a) in block form:
β_n (a) =
p_n (a) q_n (a)
r_n (a) s_n (a)
```
Here s_n (a) is a scalar and p_n (a) is of size (n − 1) × (n − 1).
We define
```
D_n (a) = det p_n (a) − I_n − 1
We regard B_n as a subgroup of B_n+1.
110
Writhe homomorphism
We define the writhe homomorphism
w: B_n → Z by w(σ_i) = 1
whenever 1 ≤ i ≤ n − 1.
111
E_n (a)
For a ∈ B_n we define
E_n (a) = (−1)^(n−1) x^(1−n−w(a) ) D_n (a).
