Intrinsic linking and knotting in straight-edge embeddings...

BackgroundProject One: K6 LinksProject Two: K7 Links

Project Three: K7 KnotsProject Four: K9

Further Work

Intrinsic linking and knotting in straight-edgeembeddings of complete graphs

Lew Ludwig

Denison UniversityGranville, Ohio

International Workshop on Spatial Graphs 2010Waseda University, Tokyo, JAPAN

August, 2010

Lew Ludwig Straight-edge links and knots

Further Work

Outline

1 Background

2 Project One: K6 Links

3 Project Two: K7 Links

4 Project Three: K7 Knots

5 Project Four: K9

6 Further Work

Further Work

Project One - Started it all. . .

1983-4: Conway and Gordon, and Sachs:K6 is intrinsically linked

Interesting side note...

Further Work

Project One - Started it all. . .

1983-4: Conway and Gordon, and Sachs:K6 is intrinsically linked

Interesting side note...

Further Work

Characterization (“Kura-cterization”)

1993: Robertson, Seymour, and Thomas:

A graph is intrinsically linked iff it contains one ofthe Petersen graphs as a minor

K6 K4,4-edgeK3,3,1

G7 G8 G9

Further Work

What next?

Examining linking and knotting in more complex orspecialized structures:

1 Every embedding contains two disjoint links

2 Links with three or more components(complexity - mnl(G ))

3 Certain types of graphs (-partite)

4 Straight-edge embeddings of graphs

Further Work

What next?

Further Work

What next?

Further Work

What next?

Further Work

What next?

Further Work

Why straight–edge embeddings?

07/24/2007 01:58 PMfig11.gif 374!362 pixels

of 1http://plus.maths.org/issue15/features/knots/fig11.gif

Polyethylene - linear/cyclic, 63 to 78 backbone atoms

Further Work

Why straight–edge embeddings?07/24/2007 01:58 PMfig11.gif 374!362 pixels

Further Work

Why straight–edge embeddings?07/24/2007 01:58 PMfig11.gif 374!362 pixels

Further Work

Project 1: The motivating question

2004: Workshop with Colin Adams

(D. Hunt, ONU)How many linked components occur in astraight–edge embedding of K6?

Recall, this number must be odd...

Further Work

Project 1: The motivating question

2004: Workshop with Colin Adams

(D. Hunt, ONU)How many linked components occur in astraight–edge embedding of K6?

Recall, this number must be odd...

Further Work

Project 1 results

(2006, Hughes)(2007, Huh and Jeon)

Every straight-edgeembedding of K6

has 1 or 3two-componentlinks

K 26 : [4,4,4,4,4,4]

Further Work

Project 1 results

K 26 : [4,4,4,4,4,4]

Further Work

Project 1 results

K 26 : [4,4,4,4,4,4]

Further Work

Project 1 results

K 26 : [4,4,4,4,4,4]

Further Work

Project 1 results

K 26 : [4,4,4,4,4,4]

Further Work

Project 1 results

(2004: Hughes and Ludwig (2006))(2007: Huh and Jeon)

K 16 : [3,3,4,4,5,5]

Further Work

Now what?

Further Work

Project 2: 2006: Arbisi and Ludwig (2010)

K7Lew Ludwig Straight-edge links and knots

Further Work

The good . . .

(3-3) links: 7

(3-4) links: 14

Further Work

The bad . . .

K27 (K3

(3-3) links: 7 or 9

(3-4) links: 14 or 18

Further Work

The ugly . . .

the INTERESTING!

Further Work

The ugly . . . the INTERESTING!

Further Work

The ugly . . . the INTERESTING!

Further Work

Counting links in K 57

Further Work

v₂17

v₂31

23(3-3) links: 13, 15, 17

(3-4) links: 26, 30, (x)(3-4) links: 26, 30, (x)

(3-4) links: 23, 27, 31

Further Work

v₂17

v₂31

(3-3) links: 13, 15, 17

(3-4) links: 26, 30, (x)(3-4) links: 26, 30, (x)

(3-4) links: 23, 27, 31

Further Work

v₂17

v₂31

(3-3) links: 13, 15, 17

(3-4) links: 26, 30, (x)

(3-4) links: 23, 27, 31

Further Work

v₂17

v₂31

23(3-3) links: 13, 15, 17

(3-4) links: 26, 30, (x)

(3-4) links: 23, 27, 31

Further Work

What next?

Examine larger structures...?

K8 has 14 distinct convex hull embeddings, eachwith a possible

)=560 (3-3) links (140)

)=280 (3-4) links (70)

)=70 (4-4) links

)=56 (5-3) links

K9 has 219 distinct convex hulls!

Further Work

What next?

)=560 (3-3) links (140)

)=280 (3-4) links (70)

)=70 (4-4) links

)=56 (5-3) links

Further Work

What next?

)=560 (3-3) links (140)

)=280 (3-4) links (70)

)=70 (4-4) links

)=56 (5-3) links

Further Work

What next?

)=560 (3-3) links (140)

)=280 (3-4) links (70)

)=70 (4-4) links

)=56 (5-3) links

Further Work

What about knots?

In 1983, Conway and Gordon also showed that K7 is intrinsicallyknotted.

For K7, how many possible knots are there?

• There are 6!/2=360 Hamiltonian cycles of length 7.

• There are 7 · 5!/2=420 Hamiltonian cycles of length 6.

Further Work

What about knots?

In 1983, Conway and Gordon also showed that K7 is intrinsicallyknotted.

For K7, how many possible knots are there?

• There are 6!/2=360 Hamiltonian cycles of length 7.

• There are 7 · 5!/2=420 Hamiltonian cycles of length 6.

Further Work

Project 3 - 2007: Grotheer and Ludwig (2009, Foisy and Ludwig)

0 14 0 18 0 17 0 24 0 30 0

1 80 0 72 0 92 0 96 0 90 0

2 164 0 174 0 143 0 123 0 120 0

3 88 1 78 1, 3 91 0, 1 90 2, 3 90 1, 2, 3, 4, 5

4 14 0 18 0, 2 16 0, 1, 2 24 0, 1 20 2, 4

5 0 0 0 0 1 0, 1 3 0 10 1, 5

6 0 0 0 0 0 0 0 0 0 0

K 17 K 2

7 K 37 K 4

7 K 57

Further WorkIn

0 14 0 18 0 17 0 24 0 30 0

1 80 0 72 0 92 0 96 0 90 0

2 164 0 174 0 143 0 123 0 120 0

3 88 1 78 1, 3 91 0, 1 90 2, 3 90 1, 2, 3, 4, 5

4 14 0 18 0, 2 16 0, 1, 2 24 0, 1 20 2, 4

5 0 0 0 0 1 0, 1 3 0 10 1, 5

6 0 0 0 0 0 0 0 0 0 0

K 17 K 2

7 K 37 K 4

7 K 57

Further Work

Project Four - 2008: Behrend and Ludwig

Recall we only looked at embeddings where all vertices were on theexternal hull: two for K6, five for K7, fourteen for K8, and so on...

Question:Given Kn with m external vertices and k = n −m internal vertices,is that embedding always ambient isotopic to an embedding with nexternal vertices?

Further Work

Project Four - 2008: Behrend and Ludwig

Recall we only looked at embeddings where all vertices were on theexternal hull: two for K6, five for K7, fourteen for K8, and so on...

Question:Given Kn with m external vertices and k = n −m internal vertices,is that embedding always ambient isotopic to an embedding with nexternal vertices?

Further Work

The idea v6

Further Work

The idea v6

Further Work

The idea v6

Further Work

The idea v6

Further Work

Further work

• Given Kn with m external vertices and k = n −m internalvertices, is that embedding always ambient isotopic to anembedding with n external vertices?

• Given Kn how many (k , m) links does it contain?3 ≤ k ≤ n − 3, 3 ≤ m ≤ n − k?

• Upper/lower bounds?

• Is every straight-edge embedding of K9 triple-linked?

Further Work

Further work

Further Work

Further work

Further Work

Further work

Further Work

Is every straight-edge embedding of K9 triple-linked?

(2001: Flapan, Naimi, and Pommershein)

K10 is intrinsically triple-linked.

Further Work

(2001: Flapan, Naimi, and Pommershein)

K10 is intrinsically triple-linked.

Further Work

K9 is NOT intrinsically triple-linked.

Further Work

Thanks. . .

• Colleen Hughes (’06)

• Pam Arbisi (’07)

• Rachel Grotheer (’08)

• Sam Berhend (’09)

• Clay Crocker and Matt Gibson (’13)

• Anderson Research Endowment

Further Work

Thanks. . .

• Colleen Hughes (’06)

• Pam Arbisi (’07)

• Rachel Grotheer (’08)

• Sam Berhend (’09)

• Clay Crocker and Matt Gibson (’13)

• Anderson Research Endowment

Intrinsic linking and knotting in straight-edge embeddings...

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