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 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 2 / 17
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
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
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 Spin c -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
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 Spin c -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
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 Spin c -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
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 Spin c -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
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 Spin c -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
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
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
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
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
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 e 1 , e 2 connecting the same pair of vertices may be redrawn as one edge of weight w ( e 1 ) + w ( e 2 ). Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17
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 e 1 , e 2 connecting the same pair of vertices may be redrawn as one edge of weight w ( e 1 ) + w ( e 2 ). 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
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 e 1 , e 2 connecting the same pair of vertices may be redrawn as one edge of weight w ( e 1 ) + w ( e 2 ). Looped edges are also allowed; a looped edge may be erased. u u+v w d d v 0 Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 4 / 17
From graphs to manifolds Example (Graphs, corresponding to some manifolds) + + − −1 0 3 2 5 −2 −2 2 − + + S 2 × S 1 S 1 × S 1 × S 1 S 3 Poincare sphere Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17
From graphs to manifolds Example (Graphs, corresponding to some manifolds) + + − −1 0 3 2 5 −2 −2 2 − + + S 2 × S 1 S 1 × S 1 × S 1 S 3 Poincare sphere Given a chainmail graph G with vertices v i and edges e ij , 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
From graphs to manifolds Example (Graphs, corresponding to some manifolds) + + − −1 0 3 2 5 −2 −2 2 − + + S 2 × S 1 S 1 × S 1 × S 1 S 3 Poincare sphere Given a chainmail graph G with vertices v i and edges e ij , i , j = 1 , 2 , . . . , n we consruct a surgery link L as follows: vertex v i → standard planar unknot L i ± 1-weighted edge e ij → ± 1-clasped ribbon linking L i and L j Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17
From graphs to manifolds Example (Graphs, corresponding to some manifolds) + + − −1 0 3 2 5 −2 −2 2 − + + S 2 × S 1 S 1 × S 1 × S 1 S 3 Poincare sphere Given a chainmail graph G with vertices v i and edges e ij , i , j = 1 , 2 , . . . , n we consruct a surgery link L as follows: vertex v i → standard planar unknot L i ± 1-weighted edge e ij → ± 1-clasped ribbon linking L i and L j + Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 5 / 17
From graphs to manifolds Linking numbers and framings of components are given by a graph Laplacian matrix Λ with entries � w ij , i � = j l ij = d ii − � n k =1 w ik , i = j Michael Polyak (Technion) From 3-manifolds to planar graphs and cycle-rooted treesNovember 27, 2014 6 / 17
From graphs to manifolds Linking numbers and framings of components are given by a graph Laplacian matrix Λ with entries � w ij , i � = j l ij = d ii − � n k =1 w ik , i = j Example (Constructing a surgery link) 5 6 4 6 1 3 1 + + + + − − 3 2 3 + 5 − 1 2 1 2 1 2 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
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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