This document presents research on classifying graphs with 7, 8, or 9 vertices as intrinsically knotted (IK), not intrinsically knotted, or indeterminate based on their embedding in 3D space. The author developed programmatic classification tests based on previous proofs to systematically classify all connected graphs on those vertex counts. The tests determine whether a graph is minor-minimal IK by checking for known IK minors, or is not IK based on properties like having fewer than 6 vertices or fewer than 15 edges. Graphs not determined by the tests are listed as indeterminate for future analysis. The goal is to further the understanding of how many minor-minimal intrinsically knotted graphs exist.

1. A Classification of All Connected Graphs
on Seven, Eight, and Nine Vertices With
Respect to the Property of Intrinsic
Knotting
Chris Morris
October 15, 2008
Chair: Dr. Tyson Henry
Member: Dr. Thomas Mattman

2. Background
2

3. What is a knot?
3

4. What is a knot?
 Exactly what you think it is!
3

5. What is a knot?
 Exactly what you think it is!
 Imagine an extension cord, tangle it, plug in the ends
3

6. What is a knot?
 Exactly what you think it is!
 Imagine an extension cord, tangle it, plug in the ends
 There is no way to ‘remove’ the knot without unplugging
the ends (or cutting the cord)
3

7. What is a knot?
 Exactly what you think it is!
 Imagine an extension cord, tangle it, plug in the ends
 There is no way to ‘remove’ the knot without unplugging
the ends (or cutting the cord)
 Can be classified, simplified and studied
3

8. What is a knot?
 Exactly what you think it is!
 Imagine an extension cord, tangle it, plug in the ends
 There is no way to ‘remove’ the knot without unplugging
the ends (or cutting the cord)
 Can be classified, simplified and studied
Unknot 3

9. What is a knot?
 Exactly what you think it is!
 Imagine an extension cord, tangle it, plug in the ends
 There is no way to ‘remove’ the knot without unplugging
the ends (or cutting the cord)
 Can be classified, simplified and studied
Unknot 3
Trefoil

10. What is a graph?
4

11. What is a graph?
 Series of vertices (points) connected by edges (lines)
4

12. What is a graph?
 Series of vertices (points) connected by edges (lines)
 Airports and flight paths
4

13. What is a graph?
 Series of vertices (points) connected by edges (lines)
 Airports and flight paths
 Connected graph: from any vertex a path exists to any
other vertex
4

14. What is a graph?
 Series of vertices (points) connected by edges (lines)
 Airports and flight paths
 Connected graph: from any vertex a path exists to any
other vertex
Not Connected
4

15. What is a graph?
 Series of vertices (points) connected by edges (lines)
 Airports and flight paths
 Connected graph: from any vertex a path exists to any
other vertex
Not Connected Connected
4

16. How do knots and graphs
relate?
5

17. How do knots and graphs
relate?
 Cycles exist in graphs which:
5

18. How do knots and graphs
relate?
 Cycles exist in graphs which:
 begin and end with same vertex
5

19. How do knots and graphs
relate?
 Cycles exist in graphs which:
 begin and end with same vertex
 travel to other vertices at most once
5

20. How do knots and graphs
relate?
 Cycles exist in graphs which:
 begin and end with same vertex
 travel to other vertices at most once
 ex. 0 ➜ 1 ➜ 3 ➜ 4 ➜ 2 ➜ 0
5

21. How do knots and graphs
relate?
0
 Cycles exist in graphs which: 4 1
 begin and end with same vertex
 travel to other vertices at most once
 ex. 0 ➜ 1 ➜ 3 ➜ 4 ➜ 2 ➜ 0 3 2
5

22. How do knots and graphs
relate?
0
 Cycles exist in graphs which: 4 1
 begin and end with same vertex
 travel to other vertices at most once
 ex. 0 ➜ 1 ➜ 3 ➜ 4 ➜ 2 ➜ 0 3 2
 Cycle is a loop, much like the extension cord
5

23. How do knots and graphs
relate?
0
 Cycles exist in graphs which: 4 1
 begin and end with same vertex
 travel to other vertices at most once
 ex. 0 ➜ 1 ➜ 3 ➜ 4 ➜ 2 ➜ 0 3 2
 Cycle is a loop, much like the extension cord
 Cycles can be knotted
5

24. What is intrinsic knotting
(IK)?
6

25. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
6

26. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
4 1
3 2
6

27. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways 2
0
4 1
3
6

28. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways 2
0
4 1
3
6

29. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
4 1
3 2
6

30. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
 Different embeddings may yield cycles with different
knots 4 1
3 2
6

31. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
 Different embeddings may yield cycles with different
knots 4 1
 Can always force a knotted embedding
3 2
6

32. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
 Different embeddings may yield cycles with different
knots 4 1
 Can always force a knotted embedding
3 2
6

33. What is intrinsic knotting
(IK)?
 Graphs can be embedded in 3 dimensional space in an
infinite number of ways
0
 Different embeddings may yield cycles with different
knots 4 1
 Can always force a knotted embedding
3 2
 Intrinsic knotting means, no matter the embedding, at least
one cycle is knotted
6

34. What is a graph minor?
7

35. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G:
7

36. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
3 2
7

37. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
3 2
7

38. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
3 2
7

39. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
7

40. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
 edge contractions
3 2
7

41. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
 edge contractions
2
7

42. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
 edge contractions
2
 G is not a minor of G
7

43. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
 edge contractions
2
 G is not a minor of G
 Minor Minimal: A property exhibited by G but not by any
of its minors
7

44. What is a graph minor?
 The graph G’ that remains after any of the following are
performed on graph G: 0
 edge removals
4 1
 vertex removals
 edge contractions
2
 G is not a minor of G
 Minor Minimal: A property exhibited by G but not by any
of its minors
 Expansion: Opposite of a minor
7

45. A Classification of All Connected Graphs
on Seven, Eight, and Nine Vertices With
Respect to the Property of Intrinsic
Knotting
8

46. Methods
9

47. What is known about intrinsic
knotting?
10

48. What is known about intrinsic
knotting?
 If H is IK and H is a minor of G, then G is IK too
10

49. What is known about intrinsic
knotting?
 If H is IK and H is a minor of G, then G is IK too
 Know that there are a finite number of minor minimal IK
graphs
10

50. What is known about intrinsic
knotting?
 If H is IK and H is a minor of G, then G is IK too
 Know that there are a finite number of minor minimal IK
graphs
 Currently about 40 are known
10

51. What is known about intrinsic
knotting?
 If H is IK and H is a minor of G, then G is IK too
 Know that there are a finite number of minor minimal IK
graphs
 Currently about 40 are known
 The big question in intrinsic knotting is: How many minor
minimal IK graphs are there total?
10

52. What is known about intrinsic
knotting?
 If H is IK and H is a minor of G, then G is IK too
 Know that there are a finite number of minor minimal IK
graphs
 Currently about 40 are known
 The big question in intrinsic knotting is: How many minor
minimal IK graphs are there total?
 Classifying graphs as IK is not easy
10

53. Why is it so difficult to
classify a graph as IK?
11

54. Why is it so difficult to
classify a graph as IK?
 Infinite number of embeddings for any graph
11

55. Why is it so difficult to
classify a graph as IK?
 Infinite number of embeddings for any graph
 If one embedding is not knotted, the graph is not IK
11

56. Why is it so difficult to
classify a graph as IK?
 Infinite number of embeddings for any graph
 If one embedding is not knotted, the graph is not IK
 No definitive approach to classify a graph as intrinsically
knotted
11

57. Why is it so difficult to
classify a graph as IK?
 Infinite number of embeddings for any graph
 If one embedding is not knotted, the graph is not IK
 No definitive approach to classify a graph as intrinsically
knotted
 Traditionally proofs are done by hand
11

58. Is this graph intrinsically
knotted?
12

59. Can we prove intrinsic
knotting?
13

60. Can we prove intrinsic
knotting?
 Proofs to show certain graphs are IK
13

61. Can we prove intrinsic
knotting?
 Proofs to show certain graphs are IK
 ex: exhibit one of the 40 as a minor
13

62. Can we prove intrinsic
knotting?
 Proofs to show certain graphs are IK
 ex: exhibit one of the 40 as a minor
 Proofs to show certain graphs are not IK
13

63. Can we prove intrinsic
knotting?
 Proofs to show certain graphs are IK
 ex: exhibit one of the 40 as a minor
 Proofs to show certain graphs are not IK
 ex: 6 vertices or less
13

64. Can we prove intrinsic
knotting?
 Proofs to show certain graphs are IK
 ex: exhibit one of the 40 as a minor
 Proofs to show certain graphs are not IK
 ex: 6 vertices or less
 No proof to show any arbitrary graph is or is not IK
13

65. What exactly did I do?
14

66. What exactly did I do?
 Classified graphs as IK, not IK or indeterminate
14

67. What exactly did I do?
 Classified graphs as IK, not IK or indeterminate
 Focused on all connected graphs on 7, 8 and 9 vertices
14

68. What exactly did I do?
 Classified graphs as IK, not IK or indeterminate
 Focused on all connected graphs on 7, 8 and 9 vertices
 Leveraged the computer to perform this classification in a
brute-force fashion
14

69. What exactly did I do?
 Classified graphs as IK, not IK or indeterminate
 Focused on all connected graphs on 7, 8 and 9 vertices
 Leveraged the computer to perform this classification in a
brute-force fashion
 Encoded proved research as programmatic classification
tests which could be applied to a graph
14

70. What exactly did I do?
 Classified graphs as IK, not IK or indeterminate
 Focused on all connected graphs on 7, 8 and 9 vertices
 Leveraged the computer to perform this classification in a
brute-force fashion
 Encoded proved research as programmatic classification
tests which could be applied to a graph
 Provided a list of indeterminate graphs which can be
scrutinized by others
14

71. The Classification Tests
15

72. The Classification Tests
 A graph is not IK if:
15

73. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
15

74. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
15

75. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
 is minor of known minor minimal IK graph
15

76. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
 is minor of known minor minimal IK graph
 has a planar subgraph after removing any two vertices
15

77. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
 is minor of known minor minimal IK graph
 has a planar subgraph after removing any two vertices
 A graph is IK if:
15

78. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
 is minor of known minor minimal IK graph
 has a planar subgraph after removing any two vertices
 A graph is IK if:
 edges ≥ (5 * vertices) – 14
15

79. The Classification Tests
 A graph is not IK if:
 vertices ≤ 6
 edges < 15
 is minor of known minor minimal IK graph
 has a planar subgraph after removing any two vertices
 A graph is IK if:
 edges ≥ (5 * vertices) – 14
 has known IK graph as a minor
15

80. The Algorithm
16

81. The Algorithm
iterate over each graph in set of graphs
16

82. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
16

83. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
apply test to graph
16

84. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
apply test to graph
done if graph is IK or not IK
16

85. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
apply test to graph
done if graph is IK or not IK
end
16

86. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
apply test to graph
done if graph is IK or not IK
end
graph is indeterminate
16

87. The Algorithm
iterate over each graph in set of graphs
iterate over each test in set of tests
apply test to graph
done if graph is IK or not IK
end
graph is indeterminate
end
16

88. The Implementation
17

89. The Implementation
 Originally implemented in Java
17

90. The Implementation
 Originally implemented in Java
 Designed algorithms for minor and planarity detection
17

91. The Implementation
 Originally implemented in Java
 Designed algorithms for minor and planarity detection
 Most ‘risky’ parts of entire design were these algorithms
17

92. The Implementation
 Originally implemented in Java
 Designed algorithms for minor and planarity detection
 Most ‘risky’ parts of entire design were these algorithms
 Wanted to use known, proven tools, instead of my
algorithms for the ‘risky’ parts
17

93. The Implementation
 Originally implemented in Java
 Designed algorithms for minor and planarity detection
 Most ‘risky’ parts of entire design were these algorithms
 Wanted to use known, proven tools, instead of my
algorithms for the ‘risky’ parts
 Transitioned to Ruby because faster interface with outside
tools
17

94. The Intrinsic Knotting Toolset
18

95. The Intrinsic Knotting Toolset
 installer
18

96. The Intrinsic Knotting Toolset
 installer
 graph_generator
18

97. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
18

98. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
 graph_complementor
18

99. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
 graph_complementor
 ik_classifier
18

100. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
 graph_complementor
 ik_classifier
 java_ik_classifier
18

101. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
 graph_complementor
 ik_classifier
 java_ik_classifier
 ik_summarizer
18

102. The Intrinsic Knotting Toolset
 installer
 graph_generator
 graph_finder
 graph_complementor
 ik_classifier
 java_ik_classifier
 ik_summarizer
 expansion_mapper
18

103. Results
19

104. 7-Vertex Graphs
20

105. 7-Vertex Graphs
 853 total connected graphs
20

106. 7-Vertex Graphs
 853 total connected graphs
 852 not intrinsically knotted
20

107. 7-Vertex Graphs
 853 total connected graphs
 852 not intrinsically knotted
 1 intrinsically knotted (K7)
20

108. 7-Vertex Graphs
 853 total connected graphs
 852 not intrinsically knotted
 1 intrinsically knotted (K7)
 0 indeterminate
20

109. 7-Vertex Graphs
 853 total connected graphs
 852 not intrinsically knotted
 1 intrinsically knotted (K7)
 0 indeterminate
 Completion Times: Java 79ms ~ Ruby 505ms
20

110. 7-Vertex Graphs
 853 total connected graphs
 852 not intrinsically knotted
 1 intrinsically knotted (K7)
 0 indeterminate
 Completion Times: Java 79ms ~ Ruby 505ms
 Max Per Graph Times: Java 1ms ~ Ruby 6ms
20

111. 8-Vertex Graphs
21

112. 8-Vertex Graphs
 11,117 total connected graphs
21

113. 8-Vertex Graphs
 11,117 total connected graphs
 11,095 not intrinsically knotted
21

114. 8-Vertex Graphs
 11,117 total connected graphs
 11,095 not intrinsically knotted
 22 intrinsically knotted
21

115. 8-Vertex Graphs
 11,117 total connected graphs
 11,095 not intrinsically knotted
 22 intrinsically knotted
 0 indeterminate
21

116. 8-Vertex Graphs
 11,117 total connected graphs
 11,095 not intrinsically knotted
 22 intrinsically knotted
 0 indeterminate
 Completion Times: Java 1.916s ~ Ruby 36.151s
21

117. 8-Vertex Graphs
 11,117 total connected graphs
 11,095 not intrinsically knotted
 22 intrinsically knotted
 0 indeterminate
 Completion Times: Java 1.916s ~ Ruby 36.151s
 Max Per Graph Times: Java 17ms ~ Ruby 2.152s
21

118. 9-Vertex Graphs
22

119. 9-Vertex Graphs
 261,080 total connected graphs
22

120. 9-Vertex Graphs
 261,080 total connected graphs
 259,055 not intrinsically knotted
22

121. 9-Vertex Graphs
 261,080 total connected graphs
 259,055 not intrinsically knotted
 1,993 intrinsically knotted
22

122. 9-Vertex Graphs
 261,080 total connected graphs
 259,055 not intrinsically knotted
 1,993 intrinsically knotted
 32 indeterminate
22

123. 9-Vertex Graphs
 261,080 total connected graphs
 259,055 not intrinsically knotted
 1,993 intrinsically knotted
 32 indeterminate
 Completion Times: Java 17m53.302s ~ Ruby 3h8m49.326s
22

124. 9-Vertex Graphs
 261,080 total connected graphs
 259,055 not intrinsically knotted
 1,993 intrinsically knotted
 32 indeterminate
 Completion Times: Java 17m53.302s ~ Ruby 3h8m49.326s
 Max Per Graph Times: Java 692ms ~ Ruby 55m8.123s
22

125. Example Indeterminate Graph
0 0
8 1 8 1
7 2
7 2
4
6 3 6 3
5 4 5
Graph 243680 Complement of 243680
23

126. Analysis & Conclusions
24

127. Classifications
25

128. Classifications
 Java and Ruby versions showed identical classification
results for every graph
25

129. Classifications
 Java and Ruby versions showed identical classification
results for every graph
 Classification which ‘determined’ IK state was useful as
‘proof’ for the classification
25

130. Classifications
 Java and Ruby versions showed identical classification
results for every graph
 Classification which ‘determined’ IK state was useful as
‘proof’ for the classification
 7-vertex classifications matched published results
25

131. Classifications
 Java and Ruby versions showed identical classification
results for every graph
 Classification which ‘determined’ IK state was useful as
‘proof’ for the classification
 7-vertex classifications matched published results
 8-vertex classifications matched published results
25

132. Classifications
 Java and Ruby versions showed identical classification
results for every graph
 Classification which ‘determined’ IK state was useful as
‘proof’ for the classification
 7-vertex classifications matched published results
 8-vertex classifications matched published results
 No published results for 9-vertex graphs for comparison,
but classifications appear realistic
25

133. Timing
26

134. Timing
 Ruby implementation ran slower than Java
26

135. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
26

136. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
 Java implementation did not degrade as much as graph
complexity increased
26

137. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
 Java implementation did not degrade as much as graph
complexity increased
 Slowest graph in Ruby took ~ 1 hour
26

138. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
 Java implementation did not degrade as much as graph
complexity increased
 Slowest graph in Ruby took ~ 1 hour
 majority of time spent in minor detection algorithm
26

139. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
 Java implementation did not degrade as much as graph
complexity increased
 Slowest graph in Ruby took ~ 1 hour
 majority of time spent in minor detection algorithm
 slowest when size difference between two graphs is greatest
26

140. Timing
 Ruby implementation ran slower than Java
 Algorithms differed, so not a ‘language comparison’
 Java implementation did not degrade as much as graph
complexity increased
 Slowest graph in Ruby took ~ 1 hour
 majority of time spent in minor detection algorithm
 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
26

141. 32 Indeterminate Graphs
27

142. 32 Indeterminate Graphs
 Potentially a new minor minimal IK graph (progress on the
‘Big Question’)
27

143. 32 Indeterminate Graphs
 Potentially a new minor minimal IK graph (progress on the
‘Big Question’)
 Left as an open area to be investigated
27

144. 32 Indeterminate Graphs
 Potentially a new minor minimal IK graph (progress on the
‘Big Question’)
 Left as an open area to be investigated
 Did discover that all of the 32 graphs arise from 5 minors
27

145. 32 Indeterminate Graphs
 Potentially a new minor minimal IK graph (progress on the
‘Big Question’)
 Left as an open area to be investigated
 Did discover that all of the 32 graphs arise from 5 minors
 Personally did not take these 32 graphs any further
27

146. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

147. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

148. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

149. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

150. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

151. 243680 245103 244632 245677 256510
243683 256338 260624
243745 245608 244064 245605 245113 244065 255925 260920 260909 260908
255244 245246 245238 256305 245239 255247 255220 260910
256368 245195 256372 256363
260922
Expansion Map of 32 Indeterminate Graphs
260928

152. Future Work
29

153. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
29

154. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
29

155. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
 Add Intrinsic Linking Classification because if a graph is not intrinsically
linked then it is not intrinsically knotted
29

156. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
 Add Intrinsic Linking Classification because if a graph is not intrinsically
linked then it is not intrinsically knotted
 Create an alternate approach to the same problem for assurance of accuracy
29

157. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
 Add Intrinsic Linking Classification because if a graph is not intrinsically
linked then it is not intrinsically knotted
 Create an alternate approach to the same problem for assurance of accuracy
 Port code to C (for increased speed)
29

158. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
 Add Intrinsic Linking Classification because if a graph is not intrinsically
linked then it is not intrinsically knotted
 Create an alternate approach to the same problem for assurance of accuracy
 Port code to C (for increased speed)
 Write code in a distributed fashion like SETI@home
29

159. Future Work
 Investigate the 32 indeterminate graphs (especially the 5 common minors)
 Investigate the Absolute Size Classification which says < 15 edges is not IK
because the smallest IK graph we found had 21 edges
 Add Intrinsic Linking Classification because if a graph is not intrinsically
linked then it is not intrinsically knotted
 Create an alternate approach to the same problem for assurance of accuracy
 Port code to C (for increased speed)
 Write code in a distributed fashion like SETI@home
 Apply tools to 10 vertex graphs and beyond
29

160. Demo
30

161. Thank You
31

162. Thank You
 Dr. Tyson Henry ~ Committee Chair
31

163. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
31

164. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
31

165. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
 Dr. Michelle Morris ~ Supportive Wife
31

166. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
 Dr. Michelle Morris ~ Supportive Wife
 Department of Computer Science
31

167. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
 Dr. Michelle Morris ~ Supportive Wife
 Department of Computer Science
 Graduate School
31

168. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
 Dr. Michelle Morris ~ Supportive Wife
 Department of Computer Science
 Graduate School
 Friends who pretended to be interested when I talked their
ears off about my project
31

169. Thank You
 Dr. Tyson Henry ~ Committee Chair
 Dr. Thomas Mattman ~ Committee Member
 Dr. Robin Soloway ~ Reviewer
 Dr. Michelle Morris ~ Supportive Wife
 Department of Computer Science
 Graduate School
 Friends who pretended to be interested when I talked their
ears off about my project
 To all of you that showed up today!
31

170. Questions
32