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From 3-manifolds to planar graphs and cycle-rooted trees Michael - - PowerPoint PPT Presentation

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From 3-manifolds to planar graphs and cycle-rooted trees Michael Polyak Technion November 27, 2014 Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 1 / 17 Michael Polyak (Technion) From

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From 3-manifolds to planar graphs and cycle-rooted trees

Michael Polyak

Technion

November 27, 2014

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 1 / 17

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Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 2 / 17

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Outline

Encode 3-manifolds by planar weighted graphs

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back Similar encodings for related objects: links in 3-manifolds, manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back Similar encodings for related objects: links in 3-manifolds, manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc. Encoding is not unique: finite set of simple moves on graphs (related to electrical networks)

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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SLIDE 7

Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back Similar encodings for related objects: links in 3-manifolds, manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc. Encoding is not unique: finite set of simple moves on graphs (related to electrical networks) Various invariants of 3-manifolds transform into combinatorial invariants

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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SLIDE 8

Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back Similar encodings for related objects: links in 3-manifolds, manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc. Encoding is not unique: finite set of simple moves on graphs (related to electrical networks) Various invariants of 3-manifolds transform into combinatorial invariants Configuration space integrals → counting of subgraphs

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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SLIDE 9

Outline

Encode 3-manifolds by planar weighted graphs Pass from various presentations of 3-manifolds to graphs and back Similar encodings for related objects: links in 3-manifolds, manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc. Encoding is not unique: finite set of simple moves on graphs (related to electrical networks) Various invariants of 3-manifolds transform into combinatorial invariants Configuration space integrals → counting of subgraphs Low-degree invariants → counting of rooted forests

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 3 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v);

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v); A vertex is balanced, if d(v) = 0 (can think about d(v) as a “defect” of v); a graph is balanced, if all of its vertices are.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v); A vertex is balanced, if d(v) = 0 (can think about d(v) as a “defect” of v); a graph is balanced, if all of its vertices are. Each edge e is decorated with a weight w(e).

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v); A vertex is balanced, if d(v) = 0 (can think about d(v) as a “defect” of v); a graph is balanced, if all of its vertices are. Each edge e is decorated with a weight w(e). A 0-weighted edge may be erased. Multiple edges are allowed. Two edges e1, e2 connecting the same pair of vertices may be redrawn as one edge of weight w(e1) + w(e2).

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v); A vertex is balanced, if d(v) = 0 (can think about d(v) as a “defect” of v); a graph is balanced, if all of its vertices are. Each edge e is decorated with a weight w(e). A 0-weighted edge may be erased. Multiple edges are allowed. Two edges e1, e2 connecting the same pair of vertices may be redrawn as one edge of weight w(e1) + w(e2). Looped edges are also allowed; a looped edge may be erased.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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SLIDE 16

Chainmail graphs

A chainmail graph is a planar graph G, decorated with Z-weights: Each vertex v is decorated with a weight d(v); A vertex is balanced, if d(v) = 0 (can think about d(v) as a “defect” of v); a graph is balanced, if all of its vertices are. Each edge e is decorated with a weight w(e). A 0-weighted edge may be erased. Multiple edges are allowed. Two edges e1, e2 connecting the same pair of vertices may be redrawn as one edge of weight w(e1) + w(e2). Looped edges are also allowed; a looped edge may be erased.

u v u+v d d w

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17

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From graphs to manifolds

Example (Graphs, corresponding to some manifolds)

−1 + + + + −2 2 − − −2 2 5 3

S3 S2 × S1 Poincare sphere S1 × S1 × S1

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17

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From graphs to manifolds

Example (Graphs, corresponding to some manifolds)

−1 + + + + −2 2 − − −2 2 5 3

S3 S2 × S1 Poincare sphere S1 × S1 × S1 Given a chainmail graph G with vertices vi and edges eij, i, j = 1, 2, . . . , n we consruct a surgery link L as follows:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17

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From graphs to manifolds

Example (Graphs, corresponding to some manifolds)

−1 + + + + −2 2 − − −2 2 5 3

S3 S2 × S1 Poincare sphere S1 × S1 × S1 Given a chainmail graph G with vertices vi and edges eij, i, j = 1, 2, . . . , n we consruct a surgery link L as follows: vertex vi → standard planar unknot Li ±1-weighted edge eij → ±1-clasped ribbon linking Li and Lj

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17

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From graphs to manifolds

Example (Graphs, corresponding to some manifolds)

−1 + + + + −2 2 − − −2 2 5 3

S3 S2 × S1 Poincare sphere S1 × S1 × S1 Given a chainmail graph G with vertices vi and edges eij, i, j = 1, 2, . . . , n we consruct a surgery link L as follows: vertex vi → standard planar unknot Li ±1-weighted edge eij → ±1-clasped ribbon linking Li and Lj

+

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17

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From graphs to manifolds

Linking numbers and framings of components are given by a graph Laplacian matrix Λ with entries lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 6 / 17

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From graphs to manifolds

Linking numbers and framings of components are given by a graph Laplacian matrix Λ with entries lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j

Example (Constructing a surgery link)

5 3 2 1 2 1 4 3 2 + 5 6 1 3 1 + 6 2 1 + − − − + +

Different graphs and surgery links for the Poincare homology sphere

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 6 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery,

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions,

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing,

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing, double covers of S3 branched along a link, etc.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing, double covers of S3 branched along a link, etc. Similar constructions work also for a variety of similar objects:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing, double covers of S3 branched along a link, etc. Similar constructions work also for a variety of similar objects: links in 3-manifolds,

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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SLIDE 31

From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing, double covers of S3 branched along a link, etc. Similar constructions work also for a variety of similar objects: links in 3-manifolds, 3-manifolds with Spin- or Spinc-structures,

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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From manifolds to graphs

It turns out, that

Theorem

Any (closed, oriented) 3-manifold can be encoded by a chainmail graph. Moreover, there are simple direct constructions starting from many different presentations of a manifold: surgery, Heegaard decompositions, plumbing, double covers of S3 branched along a link, etc. Similar constructions work also for a variety of similar objects: links in 3-manifolds, 3-manifolds with Spin- or Spinc-structures, elements of the mapping class group, etc. Some info about M can be immediately extracted from G. In particular, M is a Q-homology sphere iff det Λ = 0 and then |H1(M)| = | det Λ|; also, signature of M is the signature sign(Λ) of Λ.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 7 / 17

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Proofs and explicit constructions ... ... No time to present here.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 8 / 17

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Calculus of chainmail graphs

An encoding of a manifold by a chainmail graph is non-unique. However, there is a finite set of simple moves which allow one to pass from one chainmail graph encoding a manifold to any other graph encoding the same manifold.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 9 / 17

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Calculus of chainmail graphs

An encoding of a manifold by a chainmail graph is non-unique. However, there is a finite set of simple moves which allow one to pass from one chainmail graph encoding a manifold to any other graph encoding the same manifold. The most interesting moves are

R3 R1 R2 G G

ε −ε ε ε −ε k k+m d d −ε m + − − +

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 9 / 17

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Calculus of chainmail graphs

An encoding of a manifold by a chainmail graph is non-unique. However, there is a finite set of simple moves which allow one to pass from one chainmail graph encoding a manifold to any other graph encoding the same manifold. The most interesting moves are

R3 R1 R2 G G

ε −ε ε ε −ε k k+m d d −ε m + − − +

They are related to a number of topics: Kirby moves, relations in the mapping class group, electrical networks and cluster algebras, and Reidemeister moves for link diagrams (via balanced median graphs) -

R3 R2 R1

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 9 / 17

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Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals Rather powerful: contain universal finite type invariants of knots and 3-manifolds

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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SLIDE 40

Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals Rather powerful: contain universal finite type invariants of knots and 3-manifolds Very complicated technically

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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SLIDE 41

Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals Rather powerful: contain universal finite type invariants of knots and 3-manifolds Very complicated technically Extremely hard to compute

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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SLIDE 42

Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals Rather powerful: contain universal finite type invariants of knots and 3-manifolds Very complicated technically Extremely hard to compute We expect a similar combinatorial setup in our case: An appropriate CS-theory on graphs

discrete

− − − − − − − − − − − →

Feynman diagrams Discrete sums over subgraphs

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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SLIDE 43

Combinatorial invariants of 3-manifolds

Chern-Simons theory leads to a lot of knot and 3-manifold invariants. Attempts to understand the Jones polynomial in these terms led to quantum knot invariants, the Kontsevich integral, configuration space integrals and other constructions. In particular, Perturbative CS-theory

Feynman diagrams

− − − − − − − − − − − → Configuration space integrals Rather powerful: contain universal finite type invariants of knots and 3-manifolds Very complicated technically Extremely hard to compute We expect a similar combinatorial setup in our case: An appropriate CS-theory on graphs

discrete

− − − − − − − − − − − →

Feynman diagrams Discrete sums over subgraphs

Types of subgraphs are suggested by the theory: uni-trivalent graphs for links; trivalent graphs for 3-manifolds.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 10 / 17

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Combinatorial invariants of 3-manifolds

This actually works! Here is the setup: we pass from the manifold M to its combinatorial counter-part → a chainmail graph G. In both cases we use summations over similar Feynman graphs.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 11 / 17

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SLIDE 45

Combinatorial invariants of 3-manifolds

This actually works! Here is the setup: we pass from the manifold M to its combinatorial counter-part → a chainmail graph G. In both cases we use summations over similar Feynman graphs. Vertices of a Feynman graph: configurations of n points in M → sets of n vertices in G

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 11 / 17

slide-46
SLIDE 46

Combinatorial invariants of 3-manifolds

This actually works! Here is the setup: we pass from the manifold M to its combinatorial counter-part → a chainmail graph G. In both cases we use summations over similar Feynman graphs. Vertices of a Feynman graph: configurations of n points in M → sets of n vertices in G Edges of a Feynman graph: propagators in M → paths of edges in G

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 11 / 17

slide-47
SLIDE 47

Combinatorial invariants of 3-manifolds

This actually works! Here is the setup: we pass from the manifold M to its combinatorial counter-part → a chainmail graph G. In both cases we use summations over similar Feynman graphs. Vertices of a Feynman graph: configurations of n points in M → sets of n vertices in G Edges of a Feynman graph: propagators in M → paths of edges in G Integration over the configuration space → sum over subgraphs

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 11 / 17

slide-48
SLIDE 48

Combinatorial invariants of 3-manifolds

This actually works! Here is the setup: we pass from the manifold M to its combinatorial counter-part → a chainmail graph G. In both cases we use summations over similar Feynman graphs. Vertices of a Feynman graph: configurations of n points in M → sets of n vertices in G Edges of a Feynman graph: propagators in M → paths of edges in G Integration over the configuration space → sum over subgraphs Compactifications and anomalies due to collisions of points in M → appearance of degenerate graphs when several vertices merge together

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 11 / 17

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Θ-invariant of 3-manifolds

Let’s see this on an example of the simplest non-trivial perturbative invariant, corresponding to the Feynman graph with 2 vertices, i.e., the Θ-graph:

2 3 1 2 1

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 12 / 17

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SLIDE 50

Θ-invariant of 3-manifolds

Let’s see this on an example of the simplest non-trivial perturbative invariant, corresponding to the Feynman graph with 2 vertices, i.e., the Θ-graph:

2 3 1 2 1

We count maps φ : Θ → G with weights and multiplicities.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 12 / 17

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SLIDE 51

Θ-invariant of 3-manifolds

Let’s see this on an example of the simplest non-trivial perturbative invariant, corresponding to the Feynman graph with 2 vertices, i.e., the Θ-graph:

2 3 1 2 1

We count maps φ : Θ → G with weights and multiplicities. One can think about such a map as a choice of two vertices vi and vj of G, connected by 3 paths of edges which do not have any common internal vertices:

G

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 12 / 17

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SLIDE 52

Θ-invariant of 3-manifolds

Let’s see this on an example of the simplest non-trivial perturbative invariant, corresponding to the Feynman graph with 2 vertices, i.e., the Θ-graph:

2 3 1 2 1

We count maps φ : Θ → G with weights and multiplicities. One can think about such a map as a choice of two vertices vi and vj of G, connected by 3 paths of edges which do not have any common internal vertices:

G

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 12 / 17

slide-53
SLIDE 53

Θ-invariant of 3-manifolds

Let’s see this on an example of the simplest non-trivial perturbative invariant, corresponding to the Feynman graph with 2 vertices, i.e., the Θ-graph:

2 3 1 2 1

We count maps φ : Θ → G with weights and multiplicities. One can think about such a map as a choice of two vertices vi and vj of G, connected by 3 paths of edges which do not have any common internal vertices:

G

The weight W (φ) of φ is the product L(φ)

e∈φ(G) le, where L(φ) is the

minor of Λ, corresponding to all vertices of G not in φ(Θ).

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 12 / 17

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SLIDE 54

Θ-invariant of 3-manifolds

Degenerate maps should be counted as well. Such degeneracies appear when two vertices of the Θ-graph collide together to produce a figure-eight graph:

1 2

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 13 / 17

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SLIDE 55

Θ-invariant of 3-manifolds

Degenerate maps should be counted as well. Such degeneracies appear when two vertices of the Θ-graph collide together to produce a figure-eight graph:

1 2

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 13 / 17

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SLIDE 56

Θ-invariant of 3-manifolds

Degenerate maps should be counted as well. Such degeneracies appear when two vertices of the Θ-graph collide together to produce a figure-eight graph:

1 2

Diagonal entries of Λ also enter in the formula, when one lobe (or possibly both) of the figure-eight graph becomes a looped edge in the 4-valent

  • vertex. The weight of such a loop in vi is lii. E.g., for the map

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 13 / 17

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SLIDE 57

Θ-invariant of 3-manifolds

Degenerate maps should be counted as well. Such degeneracies appear when two vertices of the Θ-graph collide together to produce a figure-eight graph:

1 2

Diagonal entries of Λ also enter in the formula, when one lobe (or possibly both) of the figure-eight graph becomes a looped edge in the 4-valent

  • vertex. The weight of such a loop in vi is lii. E.g., for the map

k i j

we have W (φ) = L(φ) · lij · ljk · lki · lii. In the most degenerate cases – a triple edge or double looped edge – weights need to be slightly adjusted.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 13 / 17

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SLIDE 58

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 14 / 17

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SLIDE 59

Θ-invariant of 3-manifolds

Theorem

Θ(G) =

φ W (φ) is an invariant of M. If M is a Q-homology sphere

(i.e., det Λ = 0), we have Θ(G) = ±12|H1(M)|(λCW (M) − sign(M)

4

), where λCW (M) is the Casson-Walker invariant.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 15 / 17

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SLIDE 60

Θ-invariant of 3-manifolds

Theorem

Θ(G) =

φ W (φ) is an invariant of M. If M is a Q-homology sphere

(i.e., det Λ = 0), we have Θ(G) = ±12|H1(M)|(λCW (M) − sign(M)

4

), where λCW (M) is the Casson-Walker invariant.

Conjecture

The next perturbative invariant can be obtained in a similar way by counting maps of and to G. Note that Θ(G) is a polynomial of degree n + 1 in the entries of Λ. This leads to

Conjecture

Any finite type invariant of degree d of 3-manifolds (with an appropriate normalization) is a polynomial of degree at most n + d in the entries of Λ.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 15 / 17

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SLIDE 61

Θ-invariant of 3-manifolds

Remark

Instead of counting maps φ : Θ → G, we may count Θ-subgraphs of G, taking symmetries into account:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 16 / 17

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SLIDE 62

Θ-invariant of 3-manifolds

Remark

Instead of counting maps φ : Θ → G, we may count Θ-subgraphs of G, taking symmetries into account:

x2 x4 x8 x12 x6

Example

For the (negatively oriented) Poincare homology sphere one has G =

2 5 3

. Thus Λ = 1 2 2 3

  • , det Λ = −1 (so M is a Z-homology

sphere), sign(Λ) = 0, and to compute Θ(G) we count 2 · ( + ) + ( + ) + 2 · to get Θ(G) = 2· (1 · 22 + 3 · 22) + (12 + 2)(−3) + (32 + 2)(−1) + 2 · (23 − 2) = 24 and obtain λCW (M) = −2.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 16 / 17

slide-63
SLIDE 63

Cycle-rooted trees

Recall that the matrix Λ was defined as the graph Laplacian for the weight matrix W : lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 17 / 17

slide-64
SLIDE 64

Cycle-rooted trees

Recall that the matrix Λ was defined as the graph Laplacian for the weight matrix W : lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j An expression for Θ(M) in terms of the original weight matrix W (with dii

  • n the diagonal) is even simpler.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 17 / 17

slide-65
SLIDE 65

Cycle-rooted trees

Recall that the matrix Λ was defined as the graph Laplacian for the weight matrix W : lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j An expression for Θ(M) in terms of the original weight matrix W (with dii

  • n the diagonal) is even simpler.

Namely, one should count cycle-based rooted trees instead of Θ’s:

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 17 / 17

slide-66
SLIDE 66

Cycle-rooted trees

Recall that the matrix Λ was defined as the graph Laplacian for the weight matrix W : lij =

  • wij,

i = j dii − n

k=1 wik ,

i = j An expression for Θ(M) in terms of the original weight matrix W (with dii

  • n the diagonal) is even simpler.

Namely, one should count cycle-based rooted trees instead of Θ’s: Weights are defined as before, except that in the root vertex vi one uses its weight dii. No looped edges, no degenerate cases (except for a cycle being a double edge), simpler invariance check.

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 17 / 17

slide-67
SLIDE 67

Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted trees November 27, 2014 18 / 17