Intrinsic linking of n -complexes Christopher Tuffley Institute of - - PowerPoint PPT Presentation

Intrinsic linking of n-complexes

Christopher Tuffley

Institute of Fundamental Sciences Massey University, Manawatu

2010 New Zealand Mathematics Colloquium University of Otago

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 1 / 18

Outline

1

Graphs in R3 Intrinsic linking Some Ramsey-type results

2

Higher dimensions What and where Generalising the Ramsey-type results

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 2 / 18

Graphs in R3 Intrinsic linking

Intrinsic linking

Theorem (Conway and Gordon, 1983) Every embedding of the complete graph K6 in R3 contains a nontrivial link.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 3 / 18

Graphs in R3 Intrinsic linking

Intrinsic linking

Theorem (Conway and Gordon, 1983) Every embedding of the complete graph K6 in R3 contains a nontrivial link. We say that K6 is intrinsically linked.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 3 / 18

Graphs in R3 Intrinsic linking

Sketch proof

1

Define λ =

• {L,J}

link(L, J) mod 2, summing over all 1

2

6

3

• = 10 pairs of

disjoint triangles in K6.

2

λ is unchanged by ambient isotopies and crossing changes, which suffice to take any embedding to any other.

3

λ evaluates to 1 on a specific embedding. link≡ 1, link≡ 0 ⇒ Every embedding contains an odd number of links of odd linking number, hence at least one.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 4 / 18

Graphs in R3 Intrinsic linking

Sketch proof

1

Define λ =

• {L,J}

link(L, J) mod 2, summing over all 1

2

6

3

• = 10 pairs of

disjoint triangles in K6.

2

λ is unchanged by ambient isotopies and crossing changes, which suffice to take any embedding to any other.

3

λ evaluates to 1 on a specific embedding. link≡ 1, link≡ 0 ⇒ Every embedding contains an odd number of links of odd linking number, hence at least one.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 4 / 18

Graphs in R3 Intrinsic linking

Sketch proof

1

Define λ =

• {L,J}

link(L, J) mod 2, summing over all 1

2

6

3

• = 10 pairs of

disjoint triangles in K6.

2

λ is unchanged by ambient isotopies and crossing changes, which suffice to take any embedding to any other.

3

λ evaluates to 1 on a specific embedding. link≡ 0, link≡ 1 ⇒ Every embedding contains an odd number of links of odd linking number, hence at least one.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 4 / 18

Graphs in R3 Intrinsic linking

Sketch proof

1

Define λ =

• {L,J}

link(L, J) mod 2, summing over all 1

2

6

3

• = 10 pairs of

disjoint triangles in K6.

2

λ is unchanged by ambient isotopies and crossing changes, which suffice to take any embedding to any other.

3

λ evaluates to 1 on a specific embedding. link≡ 0, link≡ 1 ⇒ Every embedding contains an odd number of links of odd linking number, hence at least one.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 4 / 18

Graphs in R3 Intrinsic linking

Sketch proof

1

Define λ =

• {L,J}

link(L, J) mod 2, summing over all 1

2

6

3

• = 10 pairs of

disjoint triangles in K6.

2

λ is unchanged by ambient isotopies and crossing changes, which suffice to take any embedding to any other.

3

λ evaluates to 1 on a specific embedding. link≡ 0, link≡ 1 ⇒ Every embedding contains an odd number of links of odd linking number, hence at least one.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 4 / 18

Graphs in R3 Some Ramsey-type results

Ramsey-type results I: Necklaces and chains

Theorem (Flapan et. al., 2001 (paraphrased))

1

For N sufficiently large, every embedding of KN in R3 contains a p-component “chain”: a link L1 ∪ · · · ∪ Lp such that link(Li, Li+1) = 0 for i = 1, . . . , p − 1. (N = 6p suffices)

2

For N sufficiently large, every embedding of KN in R3 contains a p-component “necklace”: a chain such that additionally link(Lp, L1) = 0. (N = 6(p + 1) suffices)

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 5 / 18

Graphs in R3 Some Ramsey-type results

Ramsey-type results II: keyrings

Theorem (Fleming and Diesl, 2005) For N sufficiently large, every embedding of KN in R3 contains a (p + 1)-component “keyring”: a link R ∪ L1 ∪ · · · ∪ Lp such that link(R, Li) = 0 for i = 1, . . . , p. (N = O(2p) suffices)

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 6 / 18

Graphs in R3 Some Ramsey-type results

Ramsey-type results III: linking number

Theorem (Flapan, 2002) For N sufficiently large, every embedding of KN in R3 contains a two component link L ∪ J such that |link(L, J)| ≥ p. (N = p(15p − 9) suffices)

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 7 / 18

Graphs in R3 Some Ramsey-type results

Unifying principles: connect sums, additivity of link

Example (The four-to-three lemma for mod two linking number) Given a link X1 ∪ Y1 ∪ X2 ∪ Y2 in KN with link(Xi, Yi) ≡ 0 mod 2 for i = 1, 2, there is a loop X in KN with all vertices on X1 ∪ X2 such that link(X, Yi) ≡ 0 mod 2 for i = 1, 2.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 8 / 18

Graphs in R3 Some Ramsey-type results

Unifying principles: connect sums, additivity of link

Example (The four-to-three lemma for mod two linking number) Given a link X1 ∪ Y1 ∪ X2 ∪ Y2 in KN with link(Xi, Yi) ≡ 0 mod 2 for i = 1, 2, there is a loop X in KN with all vertices on X1 ∪ X2 such that link(X, Yi) ≡ 0 mod 2 for i = 1, 2.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 8 / 18

Graphs in R3 Some Ramsey-type results

Unifying principles: connect sums, additivity of link

Example (The four-to-three lemma for mod two linking number) Given a link X1 ∪ Y1 ∪ X2 ∪ Y2 in KN with link(Xi, Yi) ≡ 0 mod 2 for i = 1, 2, there is a loop X in KN with all vertices on X1 ∪ X2 such that link(X, Yi) ≡ 0 mod 2 for i = 1, 2.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 8 / 18

Graphs in R3 Some Ramsey-type results

Unifying principles: connect sums, additivity of link

Example (The four-to-three lemma for mod two linking number) Given a link X1 ∪ Y1 ∪ X2 ∪ Y2 in KN with link(Xi, Yi) ≡ 0 mod 2 for i = 1, 2, there is a loop X in KN with all vertices on X1 ∪ X2 such that link(X, Yi) ≡ 0 mod 2 for i = 1, 2.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 8 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices. Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices. Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices.

• Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices. Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices. Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

Higher dimensions: complete n-complexes

Construct K n

N, the complete n-complex on N vertices, as follows:

K 0

N: Start with N vertices.

K 1

N: Add an edge (a 1-simplex) for each pair of

vertices, to get KN. K 2

N: Add solid triangles (2-simplices) for each

triple of vertices. K 3

N: Add solid tetrahedra (3-simplices) for each

4-tuple of vertices. . . . K n

N: Add n-simplices for each (n + 1)-tuple of

vertices.

• Note that K n

n+2 ∼

= Sn.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 9 / 18

Higher dimensions What and where

The ambient space

A k- and an l-plane in Rm generically intersect in a (k + l − m)-plane.

• 2 + 2 − 3 = 1

Set k = l = n, m = 2n + 1: n + n − (2n + 1) = −1 < 0 ⇒ K n

N embeds in R2n+1.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 10 / 18

Higher dimensions What and where

The ambient space

A k- and an l-plane in Rm generically intersect in a (k + l − m)-plane.

• 2 + 2 − 3 = 1

Set k = l = n, m = 2n + 1: n + n − (2n + 1) = −1 < 0 ⇒ K n

N embeds in R2n+1.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 10 / 18

Higher dimensions What and where

Linking number

1-spheres For 1-spheres L1, L2 in R3, link(L1, L2) = the homology class [L1] in H1(R3 − L2; Z), where H1(R3 − L2; Z) ∼ = Z. n-spheres Similarly, for a n-spheres L1, L2 in R2n+1 we have Hn(R2n+1 − L2; Z) ∼ = Z so we may define link(L1, L2) = the homology class [L1] in Hn(R2n+1 − L2; Z).

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 11 / 18

Higher dimensions What and where

Linking number

1-spheres For 1-spheres L1, L2 in R3, link(L1, L2) = the homology class [L1] in H1(R3 − L2; Z), where H1(R3 − L2; Z) ∼ = Z. n-spheres Similarly, for a n-spheres L1, L2 in R2n+1 we have Hn(R2n+1 − L2; Z) ∼ = Z so we may define link(L1, L2) = the homology class [L1] in Hn(R2n+1 − L2; Z).

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 11 / 18

Higher dimensions What and where

K n

2n+4 is intrinsically linked

Theorem (Taniyama, 2000) Every embedding of K n

2n+4 in R2n+1 contains a non-trivial link.

Mimic Conway and Gordon, summing link(L1, L2) over all 1

2

2n+4

n+2

• pairs of disjoint n-spheres in K n

2n+4.

Use fact that S2n+1 is the join of two copies of Sn to construct an embedding for which the sum is 1 mod 2. Vandermonde determinants can also be used to construct a suitable embedding.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 12 / 18

Higher dimensions Generalising the Ramsey-type results

Ramsey-type results I and II: p-component links

Theorem For N sufficiently large, every embedding of K n

N in R2n+1 contains

1

a p-component “chain” (N ≥ (2n + 4)p suffices)

2

a p-component “necklace” (N ≥ (2n + 4)(p + 1) suffices)

3

a keyring with p “keys” (N = O(2p) suffices) Proofs largely unchanged from the n = 1 case. Main ingredient needed is a means to tube together spheres in K n

N.

Use known division of ∆n−1 × I into n-simplices.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 13 / 18

Higher dimensions Generalising the Ramsey-type results

Ramsey-type results III: linking number

Theorem For N sufficiently large every embedding of K n

N in R2n+1 contains a

two-component link L ∪ J such that |link(L, J)| ≥ p. For n ≥ 2 the conclusion holds for N ≥ 6(p − 1)(n + 1) 2n + 4 n + 1

• 2(p−1)n,

although the proof gives a lower but messier bound (which is unlikely to be sharp). In contrast Flapan gives N ≥ p(15p − 9) for n = 1.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 14 / 18

Higher dimensions Generalising the Ramsey-type results

Contrasting lemmas

Lemma (The four-to-three lemma for integer linking number) Suppose X1 ∪ Y1 ∪ X2 ∪ Y2 is a link in K n

N such that

link(X1, Y1) ≥ 1, link(X2, Y2) = p ≥ 1. If all components are large enough there is a link L ∪ Z ∪ W such that link(L, Z) ≥ 1, link(L, W) ≥ p, and L, Z, W are also large. n = 1: “large enough” = “also large” = “has at least q > p vertices” n ≥ 2: “large enough” = “contains a triangulated ∆n of side ℓ ≥

n

√p ” “also large” = “contains a triangulated ∆n of side ⌊nℓ/(n + 1)⌋”

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 15 / 18

Higher dimensions Generalising the Ramsey-type results

Contrasting lemmas

Lemma (The four-to-three lemma for integer linking number) Suppose X1 ∪ Y1 ∪ X2 ∪ Y2 is a link in K n

N such that

link(X1, Y1) ≥ 1, link(X2, Y2) = p ≥ 1. If all components are large enough there is a link L ∪ Z ∪ W such that link(L, Z) ≥ 1, link(L, W) ≥ p, and L, Z, W are also large. n = 1: “large enough” = “also large” = “has at least q > p vertices” n ≥ 2: “large enough” = “contains a triangulated ∆n of side ℓ ≥

n

√p ” “also large” = “contains a triangulated ∆n of side ⌊nℓ/(n + 1)⌋”

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 15 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . .

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . .

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . .

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . . . . . but n ≥ 2 requires more structure.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . . . . . but n ≥ 2 requires more structure.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

Why “large enough”?

We need spheres S1, . . . , Sk summing to zero with X1 and X2. Then 0 < p + 1 = [X1] + [X2] = −

k

• i=1

[Si] — so if k > p some [Si] must be nonnegative. For n = 1 it’s easy. . . . . . but n ≥ 2 requires more structure. A similar three-to-two lemma holds.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 16 / 18

Higher dimensions Generalising the Ramsey-type results

All over bar the induction

1

Start with p “large” copies of K n

2n+4

(all faces are triangulated with large sidelength).

2

Each contains a two-component link with nonzero linking number, by Taniyama.

3

Use the four-to-three and three-to-two lemmas to combine these

• ne by one into a single link, increasing the linking number at

each step.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 17 / 18

Higher dimensions Generalising the Ramsey-type results

Where next?

Does the following Ramsey-type result hold for n-complexes? Theorem (Flapan, Mellor and Naimi, 2008) Let p, q ∈ N. For N sufficiently large, every embedding of KN in R3 contains a p-component link L1 ∪ · · · ∪ Lp such that |link(Li, Lj)| ≥ q for all i = j.

Christopher Tuffley (Massey University) Intrinsic linking of n-complexes NZMC 2010 18 / 18