Presentation: A Classification of All Connected Graphs on Seven, Eight, and Nine Vertices With...

8/8/2019 Presentation: A Classification of All Connected Graphs on Seven, Eight, and Nine Vertices With Respect to the Property of Intrinsic Knotting

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A Classification of All Connected Graphs

on Seven, Eight, and Nine Vertices WithRespect to the Property of Intrinsic

Knotting

Chris Morris

October 15, 2008

Chair: Dr. Tyson HenryMember: Dr. Thomas Mattman


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Background

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What is a knot?

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What is a knot?

Exactly what you think it is!

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What is a knot?


Imagine an extension cord, tangle it, plug in the ends

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What is a knot?



There is no way to remove the knot without unpluggingthe ends (or cutting the cord)

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What is a knot?




Can be classified, simplified and studied

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What is a knot?





3Unknot


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What is a knot?





3Unknot Trefoil


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What is a graph?

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What is a graph?

Series of vertices (points) connected by edges (lines)

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What is a graph?


Airports and flight paths

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What is a graph?



Connected graph: from any vertex a path exists to anyother vertex

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What is a graph?




Not Connected4


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What is a graph?




Not Connected Connected4


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How do knots and graphs

relate?

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relate?

Cycles exist in graphs which:

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relate?


begin and end with same vertex

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relate?



travel to other vertices at most once

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relate?



travel to other vertices at most once ex. 0 1 3 4 2 0

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relate?




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relate?




Cycle is a loop, much like the extension cord

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d k d h


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relate?




Cycle is a loop, much like the extension cord

Cycles can be knotted

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h i i i i k i


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What is intrinsic knotting

(IK)?

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Wh i i i i k i


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(IK)?

Graphs can be embedded in 3 dimensional space in aninfinite number of ways

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Wh i i i i k i


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(IK)?


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Wh i i i i k i


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(IK)?


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Wh i i i i k i


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(IK)?


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Wh i i i i k i


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(IK)?


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Wh t i i t i i k tti


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(IK)?


Different embeddings may yield cycles with differentknots

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(IK)?



Can always force a knotted embedding

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(IK)?




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(IK)?




Intrinsic knotting means, no matter the embedding, at leastone cycle is knotted

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What is a graph minor?

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The graph G that remains after any of the following areperformed on graph G:

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edge removals

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edge removals

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edge removals vertex removals

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edge contractions

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edge contractions

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edge contractions

G is not a minor of G

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edge contractions


Minor Minimal: A property exhibited by G but not by anyof its minors

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edge contractions


Minor Minimal: A property exhibited by G but not by anyof its minors

Expansion: Opposite of a minor

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A Classification of All Connected Graphs

on Seven, Eight, and Nine Vertices WithRespect to the Property of Intrinsic

Knotting

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Methods

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What is known about intrinsic


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knotting?

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knotting?

If H is IK and H is a minor of G, then G is IK too

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knotting?


Know that there are a finite number of minor minimal IK

graphs

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knotting?



graphs

Currently about 40 are known

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knotting?



graphs


The big question in intrinsic knotting is:How many minor

minimal IK graphs are there total?

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knotting?



graphs


The big question in intrinsic knotting is:How many minor

minimal IK graphs are there total?

Classifying graphs as IK is not easy

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Why is it so difficult to


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classify a graph as IK?

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Infinite number of embeddings for any graph

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If one embedding is not knotted, the graph is notIK

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No definitive approach to classify a graph as intrinsicallyknotted

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No definitive approach to classify a graph as intrinsicallyknotted

Traditionally proofs are done by hand

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Is this graph intrinsically


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Is this graph intrinsically

knotted?

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Can we prove intrinsic


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knotting?

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knotting?

Proofs to show certain graphs are IK

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knotting?


ex: exhibit one of the 40 as a minor

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knotting?



Proofs to show certain graphs are notIK

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knotting?




ex: 6 vertices or less

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knotting?




ex: 6 vertices or less

No proof to show any arbitrary graph is or is notIK

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h l did d ?


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What exactly did I do?

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Wh l did I d ?


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Classified graphs as IK, notIK or indeterminate

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Wh l did I d ?


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Focused on all connected graphs on 7, 8 and 9 vertices

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Wh l did I d ?


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Leveraged the computer to perform this classification in abrute-force fashion

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Wh l did I d ?


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Encoded proved research as programmatic classification

tests which could be applied to a graph

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Wh l did I d ?


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Encoded proved research as programmatic classification

tests which could be applied to a graph

Provided a list of indeterminate graphs which can bescrutinized by others

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Th Cl ifi ti T t


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The Classification Tests

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Th Cl ifi ti T t


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A graph is notIK if:

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Th Cl ifi ti T t


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vertices 6

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Th Cl ifi ti T t


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vertices 6

edges < 15

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Th Cl ifi ti T t


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vertices 6

edges < 15 is minor of known minor minimal IK graph

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Th Cl ifi ti T t


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vertices 6


has a planar subgraph after removing any two vertices

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Th Cl ifi ti T t


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vertices 6



A graph is IK if:

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Th Cl ifi ti T t


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vertices 6



A graph is IK if:

edges (5 * vertices) 14

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Th Cl ifi ti T t


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vertices 6



A graph is IK if:

edges (5 * vertices) 14

has known IK graph as a minor

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Th Al ith


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The Algorithm

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The Algorithm


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The Algorithm

iterate over each graph in set of graphs

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The Algorithm


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The Algorithm


iterate over each test in set of tests

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The Algorithm


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The Algorithm



apply test to graph

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The Algorithm


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The Algorithm



apply test to graph done if graph is IK or not IK

end

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The Algorithm


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The Algorithm




end

graph is indeterminate

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The Algorithm


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The Algorithm




end

graph is indeterminate

end

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The Implementation


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The Implementation

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The Implementation


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The Implementation

Originally implemented in Java

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The Implementation


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The Implementation


Designed algorithms for minor and planarity detection

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The Implementation


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The Implementation



Most risky parts of entire design were these algorithms

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The Implementation


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The Implementation




Wanted to use known, proven tools, instead of myalgorithms for the risky parts

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The Implementation


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The Implementation




Wanted to use known, proven tools, instead of myalgorithms for the risky parts

Transitioned to Ruby because faster interface with outsidetools

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The Intrinsic Knotting Toolset


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installer

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installer

graph_generator

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installer

graph_generator

graph_finder

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installer

graph_generator

graph_finder

graph_complementor

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installer

graph_generator

graph_finder

graph_complementor

ik_classifier

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installer

graph_generator

graph_finder

graph_complementor

ik_classifier

java_ik_classifier

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installer

graph_generator

graph_finder

graph_complementor

ik_classifier

java_ik_classifier

ik_summarizer

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installer

graph_generator

graph_finder

graph_complementor

ik_classifier

java_ik_classifier

ik_summarizer

expansion_mapper

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Results

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7-Vertex Graphs


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7 Vertex Graphs

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7-Vertex Graphs


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7 Vertex Graphs

853 total connected graphs

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7-Vertex Graphs


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7 Vertex Graphs


852 notintrinsically knotted

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7-Vertex Graphs


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7 Vertex Graphs



1 intrinsically knotted (K7)

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7-Vertex Graphs


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7 Vertex Graphs




0 indeterminate

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7-Vertex Graphs


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7 Vertex Graphs




0 indeterminate

Completion Times: Java 79ms ~ Ruby 505ms

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7-Vertex Graphs


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7 Vertex Graphs




0 indeterminate

Completion Times: Java 79ms ~ Ruby 505ms

Max Per Graph Times: Java 1ms ~ Ruby 6ms

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8-Vertex Graphs


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8 Vertex Graphs

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8-Vertex Graphs


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8 Vertex Graphs

11,117 total connected graphs

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8-Vertex Graphs


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8 Vertex Graphs


11,095 notintrinsically knotted

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8-Vertex Graphs


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8 Vertex Graphs



22 intrinsically knotted

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8-Vertex Graphs


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Ve e G p s




0 indeterminate

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8-Vertex Graphs


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p




0 indeterminate

Completion Times: Java 1.916s ~ Ruby 36.151s

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8-Vertex Graphs


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p




0 indeterminate

Completion Times: Java 1.916s ~ Ruby 36.151s

Max Per Graph Times: Java 17ms ~ Ruby 2.152s

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9-Vertex Graphs


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p

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9-Vertex Graphs


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p


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9-Vertex Graphs


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p



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9-Vertex Graphs


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p



1,993 intrinsically knotted

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9-Vertex Graphs


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p




32 indeterminate

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9-Vertex Graphs


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p




32 indeterminate

Completion Times: Java 17m53.302s ~ Ruby 3h8m49.326s

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9-Vertex Graphs


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p




32 indeterminate

Completion Times: Java 17m53.302s ~ Ruby 3h8m49.326s

Max Per Graph Times: Java 692ms ~ Ruby 55m8.123s

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Example Indeterminate Graph


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p p

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Graph 243680 Complement of 243680


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Analysis & Conclusions

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Classifications


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Classifications


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Java and Ruby versions showed identical classificationresults for every graph

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Classifications


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Classification which determined IK state was useful asproof for the classification

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Classifications


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7-vertex classifications matched published results

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Classifications


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Classifications


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No published results for 9-vertex graphs for comparison,but classifications appear realistic

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Timing


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Timing


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Ruby implementation ran slower than Java

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Timing


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Algorithms differed, so not a language comparison

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Timing


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Java implementation did not degrade as much as graphcomplexity increased

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Timing


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Slowest graph in Ruby took ~ 1 hour

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Timing


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majority of time spent in minor detection algorithm

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Timing


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slowest when size difference between two graphs is greatest

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Timing


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slowest when size difference between two graphs is greatest

searching for 21 and 22 edge minors in a graph of 29 edges

on 9 vertices

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32 Indeterminate Graphs


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Potentially a new minor minimal IK graph (progress on theBig Question)

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Left as an open area to be investigated

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Did discover that all of the 32 graphs arise from 5 minors

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Did discover that all of the 32 graphs arise from 5 minors

Personally did not take these 32 graphs any further

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Expansion Map of 32 Indeterminate Graphs


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Future Work


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Future Work


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Investigate the 32 indeterminate graphs (especially the 5 common minors)

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Future Work


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Investigate the Absolute Size Classification which says < 15 edges is notIK

because the smallest IK graph we found had 21 edges

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Future Work


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Add Intrinsic Linking Classification because if a graph is notintrinsically

linked then it is notintrinsically knotted

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Future Work


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Create an alternate approach to the same problem for assurance of accuracy

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Future Work


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Port code to C (for increased speed)

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Future Work


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Port code to C (for increased speed)

Write code in a distributed fashion like SETI@home

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Demo

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Thank You


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Thank You


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Dr. Tyson Henry ~ Committee Chair

Thank You


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Dr. Thomas Mattman ~ Committee Member

Thank You


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Dr. Robin Soloway ~ Reviewer

Thank You


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Dr. Michelle Morris ~ Supportive Wife

Thank You


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Department of Computer Science

Thank You


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Graduate School

Thank You


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Graduate School

Friends who pretended to be interested when I talked theirears off about my project


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