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LINKLESS EMBEDDINGS OF GRAPHS IN 3-SPACE Robin Thomas

School of Mathematics Georgia Institute of Technology www.math.gatech.edu/˜thomas

joint work with N. Robertson and P. D. Seymour 2 Graphs are finite, undirected, may have loops and multiple edges. H is a minor of G if H can be obtained from a subgraph of G by contracting edges. 3 THEORY

All embeddings are piecewise linear 3 KNOT THEORY

All embeddings are piecewise linear

THEOREM (Papakyriakopolous) A simple closed curve in R3 is unknotted its complement has free ⇔ fundamental group. 4 lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively 4 LINKING NUMBER lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G. 4 LINKING NUMBER lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G.

THEOREM (Conway and Gordon, Sachs) K6 does not have a . 5 lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G.

THEOREM (Conway and Gordon, Sachs) K6 does not have a linkless embedding. 5 lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G.

THEOREM (Conway and Gordon, Sachs) K6 does not have a linkless embedding.

PROOF Fix embedding of K6. Consider lk(C,C0) mod 2, P sum over all unordered pairs of disjoint cycles C,C0 of G. 5 lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G.

THEOREM (Conway and Gordon, Sachs) K6 does not have a linkless embedding.

PROOF Fix embedding of K6. Consider lk(C,C0) mod 2, P sum over all unordered pairs of disjoint cycles C,C0 of G. Does not depend on the embedding • 5 lk(C,C0) = # times C goes over C0 positively # − times C goes over C0 negatively

An embedding of G in 3-space is linkless if lk(C,C0) = 0 for every two disjoint cycles C,C0 of G.

THEOREM (Conway and Gordon, Sachs) K6 does not have a linkless embedding.

PROOF Fix embedding of K6. Consider lk(C,C0) mod 2, P sum over all unordered pairs of disjoint cycles C,C0 of G. Does not depend on the embedding • = 1 for some embedding • 6 The is the set of 7 graphs that can be obtained from K6 by repeatedly applying Y ∆- and ∆Y -operations 6 The Petersen family is the set of 7 graphs that can be obtained from K6 by repeatedly applying Y ∆- and ∆Y -operations 7 THM (Sachs) No member of the Petersen family has a linkless embedding. 7 THM (Sachs) No member of the Petersen family has a linkless embedding. An embedding of a graph G in 3-space is flat if every cycle of G bounds a disk disjoint from the rest of G. 7 THM (Sachs) No member of the Petersen family has a linkless embedding. An embedding of a graph G in 3-space is flat if every cycle of G bounds a disk disjoint from the rest of G.

MAIN THEOREM THM RST A graph has a flat embedding it has no ⇔ minor isomorphic to a member of the Petersen family 7 THM (Sachs) No member of the Petersen family has a linkless embedding. An embedding of a graph G in 3-space is flat if every cycle of G bounds a disk disjoint from the rest of G.

MAIN THEOREM THM RST A graph has a flat embedding it has no ⇔ minor isomorphic to a member of the Petersen family COR it has a linkless embedding ⇔ 8 THM (B¨ohme,Saran) Let G be a flatly embedded graph, let C1,C2,...,Cn be cycles with pairwise connected intersection. Then there are disjoint open disks D1,D2,...,Dn, disjoint from the graph and such that ∂Di = Ci. 8 THM (B¨ohme,Saran) Let G be a flatly embedded graph, let C1,C2,...,Cn be cycles with pairwise connected intersection. Then there are disjoint open disks D1,D2,...,Dn, disjoint from the graph and such that ∂Di = Ci. DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. 8 THM (B¨ohme,Saran) Let G be a flatly embedded graph, let C1,C2,...,Cn be cycles with pairwise connected intersection. Then there are disjoint open disks D1,D2,...,Dn, disjoint from the graph and such that ∂Di = Ci. DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. COR Let G be planar. An embedding of G is flat it is ⇔ spherical. 9 DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. 9 DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − 9 DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − 9 DEF An embedding of a graph in R3 is spherical if the graph lies on a surface homeomorphic to S2. THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ 10 THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ 10 THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ PROOF 10 THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ 3 PROOF Let π1(R G0) be not free. − 10 THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ 3 PROOF Let π1(R G0) be not free. If e G0, then G e − 6∈ \ is not flat. 10 THM (Scharlemann, Thompson) An embedding of a graph G is spherical ⇔ (i) G is planar, and 3 (ii) π1(R G0) is free for every subgraph G0 of G − THM RST An embedding of a graph G is flat 3 ⇔ π1(R G0) is free for every subgraph G0 of G. − COR If G is embedded non-flatly then for every non-loop edge e either G e or G/e is not flat. \ 3 PROOF Let π1(R G0) be not free. If e G0, then G e − 6∈ \ is not flat. Otherwise G/e is not flat. 11 PLANAR EMBEDDINGS 11 PLANAR EMBEDDINGS

KURATOWSKI’S THEOREM A graph is planar it has ⇔ no K5 or K3,3 minor. 11 PLANAR EMBEDDINGS

KURATOWSKI’S THEOREM A graph is planar it has ⇔ no K5 or K3,3 minor. WHITNEY’S THEOREM Any two planar embeddings of the same graph differ (up to ambient isotopy) by 2-switches. 12 PLANAR EMBEDDINGS KURATOWSKI’S THEOREM A graph is planar it has ⇔ no K5 or K3,3 minor. WHITNEY’S THEOREM Any two planar embeddings of the same graph differ (up to ambient isotopy) by 2-switches. 13 PLANAR EMBEDDINGS KURATOWSKI’S THEOREM A graph is planar it has ⇔ no K5 or K3,3 minor. WHITNEY’S THEOREM Any two planar embeddings of the same graph differ (up to ambient isotopy) by 2-switches. 14 FACT Every has a unique flat embedding. 14 FACT Every planar graph has a unique flat embedding.

PROPOSITION K3,3 and K5 have two flat embeddings (up to ambient isotopy). 14 FACT Every planar graph has a unique flat embedding.

PROPOSITION K3,3 and K5 have two flat embeddings (up to ambient isotopy).

. 15 FACT Every planar graph has a unique flat embedding.

PROPOSITION K3,3 and K5 have two flat embeddings (up to ambient isotopy).

. 16 FACT Every planar graph has a unique flat embedding.

PROPOSITION K3,3 and K5 have two flat embeddings (up to ambient isotopy).

. 17 FACT Every planar graph has a unique flat embedding.

PROPOSITION K3,3 and K5 have two flat embeddings (up to ambient isotopy).

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COROLLARY A graph has a unique flat embedding it ⇔ is planar. 18 THEOREM Two nonisotopic flat embeddings differ on a K3,3 or K5 minor. 18 THEOREM Two nonisotopic flat embeddings differ on a K3,3 or K5 minor.

LEMMA In a 4-connected graph G, every two K3,3 minors “communicate”. 19 UNIQUENESS THEOREM Let φ1, φ2 be two flat embeddings of a 4-connected graph G. Then either φ1 φ2 or φ1 φ2. ' ' − 19 UNIQUENESS THEOREM Let φ1, φ2 be two flat embeddings of a 4-connected graph G. Then either φ1 φ2 or φ1 φ2. ' ' − ABOUT THE PROOF WMA G has a K3,3-minor, call it H. 19 UNIQUENESS THEOREM Let φ1, φ2 be two flat embeddings of a 4-connected graph G. Then either φ1 φ2 or φ1 φ2. ' ' − ABOUT THE PROOF WMA G has a K3,3-minor, call it H. WMA φ1 H φ2 H (by replacing φ2 by φ2 if  '  − necessary). 19 UNIQUENESS THEOREM Let φ1, φ2 be two flat embeddings of a 4-connected graph G. Then either φ1 φ2 or φ1 φ2. ' ' − ABOUT THE PROOF WMA G has a K3,3-minor, call it H. WMA φ1 H φ2 H (by replacing φ2 by φ2 if  '  − necessary). By the Lemma, φ1 H0 φ2 H0 for every  '  K3,3-minor H. 19 UNIQUENESS THEOREM Let φ1, φ2 be two flat embeddings of a 4-connected graph G. Then either φ1 φ2 or φ1 φ2. ' ' − ABOUT THE PROOF WMA G has a K3,3-minor, call it H. WMA φ1 H φ2 H (by replacing φ2 by φ2 if  '  − necessary). By the Lemma, φ1 H0 φ2 H0 for every  '  K3,3-minor H. By the previous theorem, φ1 φ2. ' 20 MAIN THEOREM

G has no minor isomorphic to a member of the Petersen family G has a flat embedding. ⇒ 21 OUTLINE OF PROOF Take a minor-minimal counterexample, G, WMA no triangles. It can be shown G is “internally 5-connected.” Take edges e = uv, f such that G/f/e, G/f e are “Kuratowski connected.” \ Let φ1 be a flat embedding of G e. \ Let φ2 be a flat embedding of G/e. Let φ3 be a flat embedding of G/f. WMA φ1/f φ3 e ' \ φ2/f φ3/e ' It can be shown that the uncontraction of f is the same in both of these embeddings. Let φ be obtained from φ3 by doing this uncontraction. Then φ e φ1, φ/e φ2, \ ' ' and so φ is flat. 22 COMMENTS

The theorem implies an O(n3) recognition • 22 COMMENTS

The theorem implies an O(n3) recognition algorithm • Not known how to verify if an embedding is linkless • 22 COMMENTS

The theorem implies an O(n3) recognition algorithm • Not known how to verify if an embedding is linkless • Is there a certifiable structure? • 22 COMMENTS

The theorem implies an O(n3) recognition algorithm • Not known how to verify if an embedding is linkless • Is there a certifiable structure? • Structure of graphs with no K6 minor? • 22 COMMENTS

The theorem implies an O(n3) recognition algorithm • Not known how to verify if an embedding is linkless • Is there a certifiable structure? • Structure of graphs with no K6 minor? •

JORGENSEN’S CONJECTURE Every 6-connected graph with no K6-minor is apex (=planar + one vertex). 23 COLIN de VERDIERE’S PARAMETER

Let µ(G) be the maximum corank of a matrix M s.t. (i) for i = j, Mij = 0 if ij E and Mij < 0 otherwise, 6 6∈ (ii) M has exactly one negative eigenvalue, (iii) if X is a symmetric n n matrix such that MX = 0 × and Xij = 0 whenever i = j or ij E, then X = 0. ∈ 23 COLIN de VERDIERE’S PARAMETER

Let µ(G) be the maximum corank of a matrix M s.t. (i) for i = j, Mij = 0 if ij E and Mij < 0 otherwise, 6 6∈ (ii) M has exactly one negative eigenvalue, (iii) if X is a symmetric n n matrix such that MX = 0 × and Xij = 0 whenever i = j or ij E, then X = 0. ∈ THM µ(G) 3 G is planar. ≤ ⇔ 23 COLIN de VERDIERE’S PARAMETER

Let µ(G) be the maximum corank of a matrix M s.t. (i) for i = j, Mij = 0 if ij E and Mij < 0 otherwise, 6 6∈ (ii) M has exactly one negative eigenvalue, (iii) if X is a symmetric n n matrix such that MX = 0 × and Xij = 0 whenever i = j or ij E, then X = 0. ∈ THM µ(G) 3 G is planar. ≤ ⇔ THM Lov´asz,Schrijver µ(G) 4 G has a flat ≤ ⇔ embedding. 24

KNOTLESS EMBEDDINGS 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. THM There exists an O(n3) algorithm to test if a graph has a knotless embedding. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. THM There exists an O(n3) algorithm to test if a graph has a knotless embedding.

PROOF Let L1,L2,... be the minor-minimal graphs that do not have knotless embeddings. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. THM There exists an O(n3) algorithm to test if a graph has a knotless embedding.

PROOF Let L1,L2,... be the minor-minimal graphs that do not have knotless embeddings. By the Theorem of Robertson and Seymour the set is finite. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. THM There exists an O(n3) algorithm to test if a graph has a knotless embedding.

PROOF Let L1,L2,... be the minor-minimal graphs that do not have knotless embeddings. By the Graph Minor Theorem of Robertson and Seymour the set is finite. 3 Whether G has an Li minor can be tested in O(n ) time by another result of Robertson and Seymour. 25 An embedding of a graph G in R3 is knotless if every cycle of G is the unknot.

THM Conway, Gordon K7 has no knotless embedding. THM There exists an O(n3) algorithm to test if a graph has a knotless embedding.

PROOF Let L1,L2,... be the minor-minimal graphs that do not have knotless embeddings. By the Graph Minor Theorem of Robertson and Seymour the set is finite. 3 Whether G has an Li minor can be tested in O(n ) time by another result of Robertson and Seymour. NOTE No explicit algorithm is known.