Introduction%20to%20Heegaard%20Floer%20Homology

1
Introduction to Heegaard Floer Homology

• Eaman Eftekhary
• Harvard University

2
General Construction

• Suppose that Y is a compact oriented
  three-manifold equipped with a self-indexing
  Morse function with a unique minimum, a unique
  maximum, g critical points of index 1 and g
  critical points of index 2.

3
General Construction

• Suppose that Y is a compact oriented
  three-manifold equipped with a self-indexing
  Morse function h with a unique minimum, a unique
  maximum, g critical points of index 1 and g
  critical points of index 2.
• The pre-image of 1.5 under h will be a surface of
  genus g which we denote by S.

4
h
R
5
Index 3 critical point
h
R
Index 0 critical point
6
Index 3 critical point
Each critical point of Index 1 or 2
will determine a curve on S
h
R
Index 0 critical point
7
Heegaard diagrams for three-manifolds

• Each critical point of index 1 or 2 determines a
  simple closed curve on the surface S. Denote the
  curves corresponding to the index 1 critical
  points by ?i, i1,,g and denote the curves
  corresponding to the index 2 critical points by
  ?i, i1,,g.

8
Heegaard diagrams for three-manifolds

• Each critical point of index 1 or 2 determines a
  simple closed curve on the surface S. Denote the
  curves corresponding to the index 1 critical
  points by ?i, i1,,g and denote the curves
  corresponding to the index 2 critical points by
  ?i, i1,,g.
• The curves ?i, i1,,g are (homologically)
  linearly independent. The same is true for ?i,
  i1,,g.

9

• We add a marked point z to the diagram, placed in
  the complement of these curves. Think of it as a
  flow line for the Morse function h, which
  connects the index 3 critical point to the index
  0 critical point.

10
The marked point z determines a flow
line connecting index-0 critical point to the
index-3 critical point
h
z
R
11

• We add a marked point z to the diagram, placed in
  the complement of these curves. Think of it as a
  flow line for the Morse function h, which
  connects the index 3 critical point to the index
  0 critical point.
• The set of data
• H(S, (?1,?2,,?g),(?1,?2,,?g),z)
• is called a pointed Heegaard diagram for the
  three-manifold Y.

12

• We add a marked point z to the diagram, placed in
  the complement of these curves. Think of it as a
  flow line for the Morse function h, which
  connects the index 3 critical point to the index
  0 critical point.
• The set of data
• H(S, (?1,?2,,?g),(?1,?2,,?g),z)
• is called a pointed Heegaard diagram for the
  three-manifold Y.
• H uniquely determines the three-manifold Y but
  not vice-versa

13
A Heegaard Diagram for S1?S2
Green curves are ??curves and the red ones
are ???curves
z
14
A different way of presenting this Heegaard
diagram
Each pair of circles of the same color
determines a handle
z
15
A different way of presenting this Heegaard
diagram
z
These arcs are completed to closed curves using
the handles
16
Knots in three-dimensional manifolds

• Any map embedding S1 in a three-manifold Y
  determines a homology class ??H1(Y,Z).

17
Knots in three-dimensional manifolds

• Any map embedding S1 to a three-manifold Y
  determines a homology class ??H1(Y,Z).
• Any such map which represents the trivial
  homology class is called a knot.

18
Knots in three-dimensional manifolds

• Any map embedding S1 to a three-manifold Y
  determines a homology class ??H1(Y,Z).
• Any such map which represents the trivial
  homology class is called a knot.
• In particular, if YS3, any embedding of S1 in S3
  will be a knot, since the first homology of S3 is
  trivial.

19
Trefoil in S3
A projection diagram for the trefoil in the
standard sphere
20
Heegaard diagrams for knots

• A pair of marked points on the surface S of a
  Heegaard diagram H for a three-manifold Y
  determine a pair of paths between the critical
  points of indices 0 and 3. These two arcs
  together determine an image of S1 embedded in Y.

21
Heegaard diagrams for knots

• A pair of marked points on the surface S of a
  Heegaard diagram H for a three-manifold Y
  determine a pair of paths between the critical
  points of indices 0 and 3. These two arcs
  together determine an image of S1 embedded in Y.
• Any knot in Y may be realized in this way using
  some Morse function and the corresponding
  Heegaard diagram.

22
Two points on the surface S determine a knot in Y
h
z
w
R
23
Heegaard diagrams for knots

• A Heegaard diagram for a knot K is a set
• H(S, (?1,?2,,?g),(?1,?2,,?g),z,w)
• where z,w are two marked points in the
  complement of the curves ?1,?2,,?g, and
  ?1,?2,,?g on the surface S.

24
Heegaard diagrams for knots

• A Heegaard diagram for a knot K is a set
  H(S, (?1,?2,,?g),(?1,?2,,?g),z,w)
• where z,w are two marked points in the
  complement of the curves ?1,?2,,?g, and
  ?1,?2,,?g on the surface S.
• There is an arc connecting z to w in the
  complement of (?1,?2,,?g), and another arc
  connecting them in the complement of
  (?1,?2,,?g). Denote them by ?? and ??.

25
Heegaard diagrams for knots

• The two marked points z,w determine the trivial
  homology class if and only if the closed curve
  ??-?? can be written as a linear combination of
  the curves (?1,?2,,?g), and (?1,?2,,?g)
  in the first homology of S.

26
Heegaard diagrams for knots

• The two marked points z,w determine the trivial
  homology class if and only if the closed curve
  ??-?? can be written as a linear combination of
  the curves (?1,?2,,?g), and (?1,?2,,?g)
  in the first homology of S.
• The first homology group of Y may be determined
  from the Heegaard diagram H
• H1(Y,Z)H1(S,Z)/?1 ?g ?1 ?g0

27
A Heegaard diagram for the trefoil
z
w
28
Constructing Heegaard diagrams for knots in S3

• Consider a plane projection of a knot K in S3.

29
Constructing Heegaard diagrams for knots in S3

• Consider a plane projection of a knot K in S3.
• Construct a surface S by thickening this
  projection.

30
Constructing Heegaard diagrams for knots in S3

• Consider a plane projection of a knot K in S3.
• Construct a surface S by thickening this
  projection.
• Construct a union of simple closed curves of two
  different colors, red and green, using the
  following procedure

31
The local construction of a Heegaard diagram from
a knot projection
32
The local construction of a Heegaard diagram from
a knot projection
33
The Heegaard diagram for trefoil after 2nd step
34
Delete the outer green curve
35
Add a new red curve and a pair of marked points
on its two sides so that the red curve
corresponds to the meridian of K.
36
The green curves denote 1st collection of simple
closed curves
The red curves denote 2nd collection of simple
closed curves
37
From topology to Heegaard diagrams

• Using this process we successfully extract a
  topological structure (a three-manifold, or a
  knot inside a three-manifold) from a set of
  combinatorial data a marked Heegaard diagram
• H(S, (?1,?2,,?g),(?1,?2,,?g),z1,,zn)
• where n is the number of marked points on
  S.

38
From Heegaard diagrams to Floer homology

• Heegaard Floer homology associates a homology
  theory to any Heegaard diagram with marked points.

39
From Heegaard diagrams to Floer homology

• Heegaard Floer homology associates a homology
  theory to any Heegaard diagram with marked
  points.
• In order to obtain an invariant of the
  topological structure, we should show that if two
  Heegaard diagrams describe the same topological
  structure (i.e. 3-manifold or knot), the
  associated homology groups are isomorphic.

40
Main construction of HFH

• Fix a Heegaard diagram
• H(S, (?1,?2,,?g),(?1,?2,,?g),z1,,zn)

41
Main construction of HFH

• Fix a Heegaard diagram
• H(S, (?1,?2,,?g),(?1,?2,,?g),z1,,zn)
• Construct the complex 2g-dimensional smooth
  manifold
• XSymg(S)(S?S??S)/S(g)
• where S(g) is the permutation group on g
  letters acting on the g-tuples of points from S.
  

42
Main construction of HFH

• Fix a Heegaard diagram
• H(S, (?1,?2,,?g),(?1,?2,,?g),z1,,zn)
• Construct the complex 2g-dimensional smooth
  manifold
• XSymg(S)(S?S??S)/S(g)
• where S(g) is the permutation group on g
  letters acting on the g-tuples of points from S.
• Every complex structure on S determines a complex
  structure on X.

43
Main construction of HFH

• Consider the two g-dimensional tori
• T??1??2 ???g and T??1??2 ???g
• in ZS?S??S. The projection map from Z to X
  embeds these two tori in X.

44
Main construction of HFH

• Consider the two g-dimensional tori
• T??1??2 ???g and T??1??2 ???g
• in ZS?S??S. The projection map from Z to X
  embeds these two tori in X.
• These tori are totally real sub-manifolds of the
  complex manifold X.

45
Main construction of HFH

• Consider the two g-dimensional tori
• T??1??2 ???g and T??1??2 ???g
• in ZS?S??S. The projection map from Z to X
  embeds these two tori in X.
• These tori are totally real sub-manifolds of the
  complex manifold X.
• If the curves ?1,?2,,?g meet the curves
  ?1,?2,,?g transversally on S, T? will meet T?
  transversally in X.

46
Intersection points of T? and T?

• A point of intersection between T? and T?
  consists of a g-tuple of points (x1,x2,,xg) such
  that for some element ??S(g) we have xi??i???(i)
  for i1,2,,g.

47
Intersection points of T? and T?

• A point of intersection between T? and T?
  consists of a g-tuple of points (x1,x2,,xg) such
  that for some element ??S(g) we have xi??i???(i)
  for i1,2,,g.
• The complex CF(H), associated with the Heegaard
  diagram H, is generated by the intersection
  points x (x1,x2,,xg) as above.
• The coefficient ring will be denoted by A,
• which is a Zu1,u2,,un-module.

48
Differential of the complex

• The differential of this complex should have the
  following form
• The values b(x,y)?A should be determined. Then
  d may be linearly extended to CF(H).

49
Differential of the complex b(x,y)

• For x,y?????? consider the space ???x,y? of the
  homotopy types of the disks satisfying the
  following properties
• u0,1?R?C?X
• u(0,t)??? , u(1,t)???
• u(s,?)x , u(s,-?)y

50
Differential of the complex b(x,y)

• For x,y?????? consider the space ???x,y? of the
  homotopy types of the disks satisfying the
  following properties
• u0,1?R?C?X
• u(0,t)??? , u(1,t)???
• u(s,?)x , u(s,-?)y
• For each ?????x,y? let M(?) denote the moduli
  space of holomorphic maps u as above representing
  the class ?.

51
Differential of the complex b(x,y)
x
u
y
X
52
Differential of the complex b(x,y)

• There is an action of R on the moduli space M(?)
  by translation of the second component by a
  constant factor If u(s,t) is holomorphic, then
  u(s,tc) is also holomorphic.

53
Differential of the complex b(x,y)

• There is an action of R on the moduli space M(?)
  by translation of the second component by a
  constant factor If u(s,t) is holomorphic, then
  u(s,tc) is also holomorphic.
• If ???? denotes the formal dimension or expected
  dimension of M(?), then the quotient moduli space
  is expected to be of dimension ????-1. We may
  manage to achieve the correct dimension.

54
Differential of the complex b(x,y)

• Let n(?? denote the number of points in the
  quotient moduli space (counted with a sign) if
  ????1. Otherwise define n(??0.

55
Differential of the complex b(x,y)

• Let n(?? denote the number of points in the
  quotient moduli space (counted with a sign) if
  ????1. Otherwise define n(??0.
• Let n(j,?? denote the intersection number
• of L(zj)zj?Symg-1(S)? Symg(S)X
• with ?.

56
Differential of the complex b(x,y)

• Let n(?? denote the number of points in the
  quotient moduli space (counted with a sign) if
  ????1. Otherwise define n(??0.
• Let n(j,?? denote the intersection number
• of L(zj)zj?Symg-1(S)? Symg(S)X
• with ?.
• Define b(x,y)?? n(??.?j uj n(j,??
• where the sum is over all ?????x,y?.

57
Two examples in dimension two
?
x
Example 1.
?
y
58
Two examples in dimension two
?
x
Example 1.
?
There is a unique holomorphic Disk, up to
reparametrization of the domain, by
Riemann Mapping theorem
y
59
Two examples in dimension two
?
x
Example 1.
?
y
d(x)y
60
Two examples in dimension two
Example 2.
x
z
y
?
w
?
61
Two examples in dimension two
There is a unique holomorphic disk from x to y,
up to reparametrization of the domain, by
Riemann Mapping theorem
Example 2.
x
z
y
?
w
?
62
Two examples in dimension two
Example 2.
x
z
y
?
w
?
63
Two examples in dimension two
Example 2.
x
z
y
The disk connecting x to z
?
w
?
64
Two examples in dimension two
Example 2.
x
z
y
?
The disk connecting z to w
w
?
65
Two examples in dimension two
Example 2.
x
z
y
?
The disk connecting y to w
w
?
66
Two examples in dimension two
Example 2.
x
z
y
?
w
?
67
Two examples in dimension two
Example 2.
x
z
y
?
There is a one parameter family of disks
connecting x to y parameterized by the length of
the cut
w
?
68
Two examples in dimension two
Example 2.
x
d(x)yz
d(y)w
d(z)-w
z
y
d(w)0
?
w
?
69
Basic properties

• The first observation is that d20.

70
Basic properties

• The first observation is that d20.
• This may be checked easily in the two examples
  discussed here.

71
Basic properties

• The first observation is that d20.
• This may be checked easily in the two examples
  discussed here.
• In general the proof uses a description of the
  boundary of M???/ when ??????. Here denotes
  the equivalence relation obtained by
  R-translation. Gromov compactness theorem and a
  gluing lemma should be used.

72
Basic properties

• Theorem (Ozsváth-Szabó) The homology groups
  HF(H,A) of the complex (CF(H),d) are invariants
  of the pointed Heegaard diagram H. For a
  three-manifold Y, or a knot (K?Y), the homology
  group is in fact independent of the specific
  Heegaard diagram used for constructing the chain
  complex and gives homology groups HF(Y,A) and
  HFK(K,A) respectively.

73
Refinements of these homology groups

• Consider the space Spinc(Y) of Spinc-structures
  on Y. This is the space of homology classes of
  nowhere vanishing vector fields on Y. Two
  non-vanishing vector fields on Y are called
  homologous if they are isotopic in the complement
  of a ball in Y.

74
Refinements of these homology groups

• Consider the space Spinc(Y) of Spinc-structures
  on Y. This is the space of homology classes of
  nowhere vanishing vector fields on Y. Two
  non-vanishing vector fields on Y are called
  homologous if they are isotopic in the complement
  of a ball in Y.
• The marked point z defines a map sz from the set
  of generators of CF(H) to Spinc(Y)
• sz??????Spinc(Y)
• defined as follows

75
Refinements of these homology groups

• If x(x1,x2,,xg)?????? is an intersection
  point, then each of xj determines a flow line for
  the Morse function h connecting one of the
  index-1 critical points to an index-2 critical
  point. The marked point z determines a flow line
  connecting the index-0 critical point to the
  index-3 critical point.

76
Refinements of these homology groups

• If x(x1,x2,,xg)?????? is an intersection
  point, then each of xj determines a flow line for
  the Morse function h connecting one of the
  index-1 critical points to an index-2 critical
  point. The marked point z determines a flow line
  connecting the index-0 critical point to the
  index-3 critical point.
• All together we obtain a union of flow lines
  joining pairs of critical points of indices of
  different parity.

77
Refinements of these homology groups

• The gradient vector field may be modified in a
  neighborhood of these paths to obtain a nowhere
  vanishing vector field on Y.

78
Refinements of these homology groups

• The gradient vector field may be modified in a
  neighborhood of these paths to obtain a nowhere
  vanishing vector field on Y.
• The class of this vector field in Spinc(Y) is
  independent of this modification and is denoted
  by sz(x).

79
Refinements of these homology groups

• The gradient vector field may be modified in a
  neighborhood of these paths to obtain a nowhere
  vanishing vector field on Y.
• The class of this vector field in Spinc(Y) is
  independent of this modification and is denoted
  by sz(x).
• If x,y?????? are intersection points with
• ???x,y???, then sz(x) sz(y).

80
Refinements of these homology groups

• This implies that the homology groups HF(Y,A)
  decompose according to the Spinc structures over
  Y
• HF(Y,A)?s?Spin(Y)HF(Y,As)

81
Refinements of these homology groups

• This implies that the homology groups HF(Y,A)
  decompose according to the Spinc structures over
  Y
• HF(Y,A)?s?Spin(Y)HF(Y,As)
• For each s?Spinc(Y) the group HF(Y,As) is also
  an invariant of the three-manifold Y and the
  Spinc structure s.

82
Some examples

• For S3, Spinc(S3)s0 and HF(Y,As0)A

83
Some examples

• For S3, Spinc(S3)s0 and HF(Y,As0)A
• For S1?S2, Spinc(S1?S2)Z. Let s0 be the Spinc
  structure such that c1(s0)0, then for s?s0,
  HF(Y,As)0. Furthermore we have HF(Y,As0)A?A,
  where the homological gradings of the two copies
  of A differ by 1.

84
Heegaard diagram for S3
z
The opposite sides of the rectangle should be
identified to obtain a torus (surface of genus1)
x
85
Heegaard diagram for S3
z
Only one generator x, and no differentials so
the homology will be A
x
86
Heegaard diagram for S1?S2
z
Only two generators x,y and two homotopy classes
of disks of index 1.
x
y
87
Heegaard diagram for S1?S2
z
The first disk connecting x to y, with Maslov
index one.
x
y
88
Heegaard diagram for S1?S2
z
The second disk connecting x to y, with Maslov
index one. The sign will be different from the
first one.
x
y
89
Heegaard diagram for S1?S2
z
d(x)d(y)0 sz(x)sz(y)s0 ?(x)?(y)11 HF(S1?S2,
A,s0) A?x??A?y?
x
y
90
Some other simple cases

• Lens spaces L(p,q)

91
Some other simple cases

• Lens spaces L(p,q)
• S3n(K) the result of n-surgery on alternating
  knots in S3. The result may be understood in
  terms of the Alexander polynomial of the knot.

92
Some other simple cases

• Lens spaces L(p,q)
• S3n(K) the result of n-surgery on alternating
  knots in S3. The result may be understood in
  terms of the Alexander polynomial of the knot.
• Connected sums of pieces of the above type There
  is a connected sum formula.

93
Connected sum formula

• Spinc(Y1Y2)Spinc(Y1)?Spinc(Y2) Maybe the
  better notation is Spinc(Y1Y2)Spinc(Y1)Spinc(Y2
  )

94
Connected sum formula

• Spinc(Y1Y2)Spinc(Y1)?Spinc(Y2) Maybe the
  better notation is Spinc(Y1Y2)Spinc(Y1)Spinc(Y2
  )
• HF(Y1Y2,As1s2)
• HF(Y1,As1)?AHF(Y2,As2)

95
Connected sum formula

• Spinc(Y1Y2)Spinc(Y1)?Spinc(Y2) Maybe the
  better notation is Spinc(Y1Y2)Spinc(Y1)Spinc(Y2
  )
• HF(Y1Y2,As1s2)
• HF(Y1,As1)?AHF(Y2,As2)
• In particular for AZ, as a trivial Zu1-module,
  the connected sum formula is usually simple (in
  practice).

96
Refinements for knots

• Consider the space of relative Spinc structures
  Spinc(Y,K) for a knot (Y,K)

97
Refinements for knots

• Consider the space of relative Spinc structures
  Spinc(Y,K) for a knot (Y,K)
• Spinc(Y,K) is by definition the space of homology
  classes of non-vanishing vector fields in the
  complement of K which converge to the orientation
  of K.

98
Refinements for knots

• The pair of marked points (z,w) on a Heegaard
  diagram H for K determine a map from the set of
  generators x?????? to Spinc(Y,K),
  denoted by sK(x) ?Spinc(Y,K).

99
Refinements for knots

• The pair of marked points (z,w) on a Heegaard
  diagram H for K determine a map from the set of
  generators x?????? to Spinc(Y,K),
  denoted by sK(x) ?Spinc(Y,K).
• In the simplest case where AZ, the coefficient
  of any y?????? in d(x) is zero, unless
  sK(x)sK(y).

100
Refinements for knots

• This is a better refinement in comparison with
  the previous one for three-manifolds
• Spinc(Y,K)Z?Spinc(Y)

101
Refinements for knots

• This is a better refinement in comparison with
  the previous one for three-manifolds
• Spinc(Y,K)Z?Spinc(Y)
• In particular for YS3 and standard knots we have
  
• Spinc(K)Spinc(S3,K)Z
• We restrict ourselves to this case, with AZ!
  

102
Some results for knots in S3

• For each s?Z, we obtain a homology group HF(K,s)
  which is an invariant for K.

103
Some results for knots in S3

• For each s?Z, we obtain a homology group HF(K,s)
  which is an invariant for K.
• There is a homological grading induced on
  HF(K,s). As a result
• HF(K,s)?i?Z HFi(K,s)

104
Some results for knots in S3

• For each s?Z, we obtain a homology group HF(K,s)
  which is an invariant for K.
• There is a homological grading induced on
  HF(K,s). As a result
• HF(K,s)?i?Z HFi(K,s)
• So each HF(K,s) has a well-defined Euler
  characteristic ?(K,s)

105
Some results for knots in S3

• The polynomial
• PK(t)?s?Z ?(K,s).ts
• will be the symmetrized Alexander polynomial
  of K.

106
Some results for knots in S3

• The polynomial
• PK(t)?s?Z ?(K,s).ts
• will be the symmetrized Alexander polynomial
  of K.
• There is a symmetry as follows
• HFi(K,s)HFi-2s(K,-s)

107
Some results for knots in S3

• The polynomial
• PK(t)?s?Z ?(K,s).ts
• will be the symmetrized Alexander polynomial
  of K.
• There is a symmetry as follows
• HFi(K,s)HFi-2s(K,-s)
• HF(K) determines the genus of K as follows

108
Genus of a knot

• Suppose that K is a knot in S3.

109
Genus of a knot

• Suppose that K is a knot in S3.
• Consider all the oriented surfaces C with one
  boundary component in S3\K such that the boundary
  of C is K.

110
Genus of a knot

• Suppose that K is a knot in S3.
• Consider all the oriented surfaces C with one
  boundary component in S3\K such that the boundary
  of C is K.
• Such a surface is called a Seifert surface for K.

111
Genus of a knot

• Suppose that K is a knot in S3.
• Consider all the oriented surfaces C with one
  boundary component in S3\K such that the boundary
  of C is K.
• Such a surface is called a Seifert surface for K.
• The genus g(K) of K is the minimum genus for a
  Seifert surface for K.

112
HFH determines the genus

• Let d(K) be the largest integer s such that
  HF(K,s) is non-trivial.

113
HFH determines the genus

• Let d(K) be the largest integer s such that
  HF(K,s) is non-trivial.
• Theorem (Ozsváth-Szabó) For any knot K in S3,
  d(K)g(K).

114
HFH and the 4-ball genus

• In fact there is a slightly more interesting
  invariant ?(K) defined from HF(K,A), where
  AZu1-1,u2-1, which gives a lower bound for the
  4-ball genus g4(K) of K.

115
HFH and the 4-ball genus

• In fact there is a slightly more interesting
  invariant ?(K) defined from HF(K,A), where
  AZu1-1,u2-1, which gives a lower bound for the
  4-ball genus g4(K) of K.
• The 4-ball genus in the smallest genus of a
  surface in the 4-ball with boundary K in S3,
  which is the boundary of the 4-ball.

116
HFH and the 4-ball genus

• The 4-ball genus gives a lower bound for the
  un-knotting number u(K) of K.

117
HFH and the 4-ball genus

• The 4-ball genus gives a lower bound for the
  un-knotting number u(K) of K.
• Theorem(Ozsváth-Szabó)
• ?(K) g4(K)u(K)

118
HFH and the 4-ball genus

• The 4-ball genus gives a lower bound for the
  un-knotting number u(K) of K.
• Theorem(Ozsváth-Szabó)
• ?(K) g4(K)u(K)
• Corollary(Milnor conjecture, 1st proved by
  Kronheimer-Mrowka using gauge theory)
• If T(p,q) denotes the (p,q) torus knot, then
  u(T(p,q))(p-1)(q-1)/2

119
T(p,q) p strands, q twists
120
Compations

• HF(K) is completely determined from the
  symmetrized Alexander polynomial and the
  signature s(K), if K is an alternating knot.

121
Compations

• HF(K) is completely determined from the
  symmetrized Alexander polynomial and the
  signature s(K), if K is an alternating knot.
• Torus knots, three-strand pretzel knots, etc.

122
Compations

• HF(K) is completely determined from the
  symmetrized Alexander polynomial and the
  signature s(K), if K is an alternating knot.
• Torus knots, three-strand pretzel knots, etc.
• Small knots We know the answer for all knots up
  to 14 crossings.

123
Why is it possible to compute?

• There is an easy way to understand the homotopy
  classes of disks in ???x,y) when the associated
  relative Spinc structures associated with x,y in
  Spinc(K) are the same.

124
Why is it possible to compute?

• There is an easy way to understand the homotopy
  classes of disks in ???x,y) when the associated
  relative Spinc structures associated with x,y in
  Spinc(K) are the same.
• Let ? be an element in ???x,y), and let
  z1,z2,,zm be marked points on S, one in each
  connected component of the complement of the
  curves in S.

125
Why is it possible to compute?

• Consider the subspaces L(zj)zj?Symg-1(S) and
  let n(j,?) be the intersection number of ? with
  L(zj).

126
Why is it possible to compute?

• Consider the subspaces L(zj)zj?Symg-1(S) and
  let n(j,?) be the intersection number of ? with
  L(zj).
• The collection of integers n(j,?), j1,,m
  determine the homotopy class ?.

127
Why is it possible to compute?

• Consider the subspaces L(zj)zj?Symg-1(S) and
  let n(j,?) be the intersection number of ? with
  L(zj).
• The collection of integers n(j,?), j1,,m
  determine the homotopy class ?.
• There is a simple combinatorial way to check if
  such a collection determines a homotopy class in
  ???x,y) or not.

128
Why is it possible to compute?

• There is a combinatorial formula for the expected
  dimension of ???? of M(?) in terms of n(j,?) and
  the geometry of the curves on S.

129
Why is it possible to compute?

• There is a combinatorial formula for the expected
  dimension of ???? of M(?) in terms of n(j,?) and
  the geometry of the curves on S.
• We know that if n(?) is not zero, then ????1,
  and all n(j,?) are non-negative. Furthermore, if
  zz1 and wz2, then n(1,?) n(2,?)0.

130
Why is it possible to compute?

• These are strong restrictions. For example these
  restrictions are enough for a complete
  computation for alternating knots.

131
Why is it possible to compute?

• These are strong restrictions. For example these
  restrictions are enough for a complete
  computation for alternating knots.
• In other cases, these are still pretty strong,
  and help a lot with the computations.

132
Why is it possible to compute?

• These are strong restrictions. For example these
  restrictions are enough for a complete
  computation for alternating knots.
• In other cases, these are still pretty strong,
  and help a lot with the computations.
• There are computer programs (e.g. by Monalescue)
  which provide all the simplifications of the
  above type in the computations.

133
Some domains for which the moduli space is known
x
Any 2n-gone as shown here with alternating red
and green edges corresponds to as moduli space
contributing 1 to the differential
y
y
x
y
x
134
Some domains for which the moduli space is known
x
The same is true for the same type of polygons
with a number of circles excluded as shown in
the picture.
y
xy
y
x
y
x
135
Relation to the three-manifold invariants

• Theorem (Ozsváth-Szabó) Heegaard Floer complex
  for a knot K determines the Heegaard Floer
  homology for three-manifolds obtained by surgery
  on K.

136
Relation to the three-manifold invariants

• Theorem (E.) More generally if a 3-manifold is
  obtained from two knot-complements by identifying
  them on the boundary, then the Heegaard Floer
  complexs of the two knots, determine the Heegaard
  Floer homology of the resulting three-manifold