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 Graphs in R 3 1 Intrinsic linking Some Ramsey-type results Higher dimensions 2 What and where Generalising the Ramsey-type results Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 2 / 18
Graphs in R 3 Intrinsic linking Intrinsic linking Theorem (Conway and Gordon, 1983) Every embedding of the complete graph K 6 in R 3 contains a nontrivial link. Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 3 / 18
Graphs in R 3 Intrinsic linking Intrinsic linking Theorem (Conway and Gordon, 1983) Every embedding of the complete graph K 6 in R 3 contains a nontrivial link. We say that K 6 is intrinsically linked . Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 3 / 18
Graphs in R 3 Intrinsic linking Sketch proof Define 1 � λ = link ( L , J ) mod 2 , { L , J } � 6 summing over all 1 � = 10 pairs of 2 3 disjoint triangles in K 6 . λ is unchanged by ambient isotopies 2 and crossing changes, which suffice to take any embedding to any other. λ evaluates to 1 on a specific 3 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 R 3 Intrinsic linking Sketch proof Define 1 � λ = link ( L , J ) mod 2 , { L , J } � 6 summing over all 1 � = 10 pairs of 2 3 disjoint triangles in K 6 . λ is unchanged by ambient isotopies 2 and crossing changes, which suffice to take any embedding to any other. λ evaluates to 1 on a specific 3 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 R 3 Intrinsic linking Sketch proof Define 1 � λ = link ( L , J ) mod 2 , { L , J } � 6 summing over all 1 � = 10 pairs of 2 3 disjoint triangles in K 6 . λ is unchanged by ambient isotopies 2 and crossing changes, which suffice to take any embedding to any other. λ evaluates to 1 on a specific 3 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 R 3 Intrinsic linking Sketch proof Define 1 � λ = link ( L , J ) mod 2 , { L , J } � 6 summing over all 1 � = 10 pairs of 2 3 disjoint triangles in K 6 . λ is unchanged by ambient isotopies 2 and crossing changes, which suffice to take any embedding to any other. λ evaluates to 1 on a specific 3 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 R 3 Intrinsic linking Sketch proof Define 1 � λ = link ( L , J ) mod 2 , { L , J } � 6 summing over all 1 � = 10 pairs of 2 3 disjoint triangles in K 6 . λ is unchanged by ambient isotopies 2 and crossing changes, which suffice to take any embedding to any other. λ evaluates to 1 on a specific 3 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 R 3 Some Ramsey-type results Ramsey-type results I: Necklaces and chains Theorem (Flapan et. al. , 2001 (paraphrased)) For N sufficiently large, every embedding of K N in R 3 contains a 1 p-component “chain”: a link L 1 ∪ · · · ∪ L p such that link ( L i , L i + 1 ) � = 0 for i = 1 , . . . , p − 1 . (N = 6 p suffices) For N sufficiently large, every embedding of K N in R 3 contains a 2 p-component “necklace”: a chain such that additionally link ( L p , L 1 ) � = 0 . (N = 6 ( p + 1 ) suffices) Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 5 / 18
Graphs in R 3 Some Ramsey-type results Ramsey-type results II: keyrings Theorem (Fleming and Diesl, 2005) For N sufficiently large, every embedding of K N in R 3 contains a ( p + 1 ) -component “keyring”: a link R ∪ L 1 ∪ · · · ∪ L p such that link ( R , L i ) � = 0 (N = O ( 2 p ) suffices) for i = 1 , . . . , p. Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 6 / 18
Graphs in R 3 Some Ramsey-type results Ramsey-type results III: linking number Theorem (Flapan, 2002) For N sufficiently large, every embedding of K N in R 3 contains a two component link L ∪ J such that | link ( L , J ) | ≥ p . (N = p ( 15 p − 9 ) suffices) Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 7 / 18
Graphs in R 3 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 X 1 ∪ Y 1 ∪ X 2 ∪ Y 2 in K N with link ( X i , Y i ) �≡ 0 mod 2 for i = 1 , 2, there is a loop X in K N with all vertices on X 1 ∪ X 2 such that link ( X , Y i ) �≡ 0 mod 2 for i = 1 , 2. Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 8 / 18
Graphs in R 3 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 X 1 ∪ Y 1 ∪ X 2 ∪ Y 2 in K N with link ( X i , Y i ) �≡ 0 mod 2 for i = 1 , 2, there is a loop X in K N with all vertices on X 1 ∪ X 2 such that link ( X , Y i ) �≡ 0 mod 2 for i = 1 , 2. Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 8 / 18
Graphs in R 3 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 X 1 ∪ Y 1 ∪ X 2 ∪ Y 2 in K N with link ( X i , Y i ) �≡ 0 mod 2 for i = 1 , 2, there is a loop X in K N with all vertices on X 1 ∪ X 2 such that link ( X , Y i ) �≡ 0 mod 2 for i = 1 , 2. Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 8 / 18
Graphs in R 3 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 X 1 ∪ Y 1 ∪ X 2 ∪ Y 2 in K N with link ( X i , Y i ) �≡ 0 mod 2 for i = 1 , 2, there is a loop X in K N with all vertices on X 1 ∪ X 2 such that link ( X , Y i ) �≡ 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 K N . 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. n + 2 ∼ Note that K n = S n . 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 K N . 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. n + 2 ∼ Note that K n = S n . Christopher Tuffley (Massey University) Intrinsic linking of n -complexes NZMC 2010 9 / 18
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