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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
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.
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KNOT THEORY
All embeddings are piecewise linear
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KNOT THEORY
All embeddings are piecewise linear
THEOREM (Papakyriakopolous) A simple closed curve
in R3 is unknotted ⇔ its complement has free
fundamental group.
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LINKING NUMBER
lk(C, C 0) = # times C goes over C 0 positively − #
times C goes over C 0 negatively
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LINKING NUMBER
lk(C, C 0) = # times C goes over C 0 positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
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LINKING NUMBER
lk(C, C 0) = # times C goes over C 0 positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
THEOREM (Conway and Gordon, Sachs) K6 does not
have a linkless embedding.
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0
0
lk(C, C ) = # times C goes over C positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
THEOREM (Conway and Gordon, Sachs) K6 does not
have a linkless embedding.
5
0
0
lk(C, C ) = # times C goes over C positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
THEOREM (Conway and Gordon, Sachs) K6 does not
have a linkless embedding.
PROOF Fix embedding of K6. Consider
P
lk(C, C 0) mod 2,
sum over all unordered pairs of disjoint cycles C, C 0 of G.
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0
0
lk(C, C ) = # times C goes over C positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
THEOREM (Conway and Gordon, Sachs) K6 does not
have a linkless embedding.
PROOF Fix embedding of K6. Consider
P
lk(C, C 0) mod 2,
sum over all unordered pairs of disjoint cycles C, C 0 of G.
• Does not depend on the embedding
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0
0
lk(C, C ) = # times C goes over C positively − #
times C goes over C 0 negatively
An embedding of G in 3-space is linkless if lk(C, C 0) = 0
for every two disjoint cycles C, C 0 of G.
THEOREM (Conway and Gordon, Sachs) K6 does not
have a linkless embedding.
PROOF Fix embedding of K6. Consider
P
lk(C, C 0) mod 2,
sum over all unordered pairs of disjoint cycles C, C 0 of G.
• Does not depend on the embedding
• = 1 for some embedding
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The Petersen family is the set of 7 graphs that can be
obtained from K6 by repeatedly applying Y ∆- and
∆Y -operations
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The Petersen family is the set of 7 graphs that can be
obtained from K6 by repeatedly applying Y ∆- and
∆Y -operations
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THM (Sachs) No member of the Petersen family has a
linkless embedding.
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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.
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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
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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
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THM (Böhme, 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.
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THM (Böhme, 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 S 2.
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THM (Böhme, 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 S 2.
COR Let G be planar. An embedding of G is flat ⇔ it is
spherical.
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3
DEF An embedding of a graph in R is spherical if the
graph lies on a surface homeomorphic to S 2.
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3
DEF An embedding of a graph in R is spherical if the
graph lies on a surface homeomorphic to S 2.
THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
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3
DEF An embedding of a graph in R is spherical if the
graph lies on a surface homeomorphic to S 2.
THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − G0) is free for every subgraph G0 of G.
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3
DEF An embedding of a graph in R is spherical if the
graph lies on a surface homeomorphic to S 2.
THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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.
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THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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.
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THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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
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THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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 Let π1(R3 − G0) be not free.
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THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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 Let π1(R3 − G0) be not free. If e 6∈ G0, then G\e
is not flat.
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THM (Scharlemann, Thompson) An embedding of a
graph G is spherical ⇔
(i) G is planar, and
(ii) π1(R3 − G0) is free for every subgraph G0 of G
THM RST An embedding of a graph G is flat ⇔
π1(R3 − 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 Let π1(R3 − G0) be not free. If e 6∈ G0, then G\e
is not flat. Otherwise G/e is not flat.
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PLANAR EMBEDDINGS
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PLANAR EMBEDDINGS
KURATOWSKI’S THEOREM A graph is planar ⇔ it has
no K5 or K3,3 minor.
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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.
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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.
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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.
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FACT Every planar graph has a unique flat embedding.
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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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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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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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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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FACT Every planar graph has a unique flat embedding.
PROPOSITION K3,3 and K5 have two flat embeddings
(up to ambient isotopy).
.
COROLLARY A graph has a unique flat embedding ⇔ it
is planar.
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THEOREM Two nonisotopic flat embeddings differ on a
K3,3 or K5 minor.
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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”.
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UNIQUENESS THEOREM Let φ1, φ2 be two flat
embeddings of a 4-connected graph G. Then either
φ1 ' φ2 or φ1 ' −φ2.
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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.
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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).
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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 H 0 ' φ2 H 0 for every
K3,3-minor H.
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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 H 0 ' φ2 H 0 for every
K3,3-minor H. By the previous theorem, φ1 ' φ2.
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MAIN THEOREM
G has no minor isomorphic to a member of the Petersen
family ⇒ G has a flat embedding.
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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.
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COMMENTS
• The theorem implies an O(n3) recognition algorithm
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COMMENTS
• The theorem implies an O(n3) recognition algorithm
• Not known how to verify if an embedding is linkless
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COMMENTS
• The theorem implies an O(n3) recognition algorithm
• Not known how to verify if an embedding is linkless
• Is there a certifiable structure?
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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?
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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).
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COLIN de VERDIERE’S PARAMETER
Let µ(G) be the maximum corank of a matrix M s.t.
(i) for i 6= j, Mij = 0 if ij 6∈ E and Mij < 0 otherwise,
(ii) M has exactly one negative eigenvalue,
(iii) if X is a symmetric n × n matrix such that M X = 0
and Xij = 0 whenever i = j or ij ∈ E, then X = 0.
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COLIN de VERDIERE’S PARAMETER
Let µ(G) be the maximum corank of a matrix M s.t.
(i) for i 6= j, Mij = 0 if ij 6∈ E and Mij < 0 otherwise,
(ii) M has exactly one negative eigenvalue,
(iii) if X is a symmetric n × n matrix such that M X = 0
and Xij = 0 whenever i = j or ij ∈ E, then X = 0.
THM µ(G) ≤ 3 ⇔ G is planar.
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COLIN de VERDIERE’S PARAMETER
Let µ(G) be the maximum corank of a matrix M s.t.
(i) for i 6= j, Mij = 0 if ij 6∈ E and Mij < 0 otherwise,
(ii) M has exactly one negative eigenvalue,
(iii) if X is a symmetric n × n matrix such that M X = 0
and Xij = 0 whenever i = j or ij ∈ E, then X = 0.
THM µ(G) ≤ 3 ⇔ G is planar.
THM Lovász, Schrijver µ(G) ≤ 4 ⇔ G has a flat
embedding.
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KNOTLESS EMBEDDINGS
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An embedding of a graph G in R is knotless if every
cycle of G is the unknot.
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3
An embedding of a graph G in R is knotless if every
cycle of G is the unknot.
THM Conway, Gordon K7 has no knotless embedding.
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3
An embedding of a graph G in R 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.
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3
An embedding of a graph G in R 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.
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3
An embedding of a graph G in R 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.
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3
An embedding of a graph G in R 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.
Whether G has an Li minor can be tested in O(n3) time
by another result of Robertson and Seymour.
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3
An embedding of a graph G in R 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.
Whether G has an Li minor can be tested in O(n3) time
by another result of Robertson and Seymour.
NOTE No explicit algorithm is known.
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