SLIDE 1
A Tale of Knots & Games
Allison Henrich, Ph.D.
Seattle University
April 27, 2014
Allison Henrich, Ph.D. A Tale of Knots & Games
What is a knot?
Allison Henrich, Ph.D. A Tale of Knots & Games
A Tale of Knots & Games Allison Henrich, Ph.D. Seattle - - PowerPoint PPT Presentation
A Tale of Knots & Games Allison Henrich, Ph.D. Seattle - - PowerPoint PPT Presentation
A Tale of Knots & Games Allison Henrich, Ph.D. Seattle University April 27, 2014 Allison Henrich, Ph.D. A Tale of Knots & Games What is a knot? Allison Henrich, Ph.D. A Tale of Knots & Games What is a knot? (Well come back
SLIDE 2
SLIDE 3
What is a knot?
(We’ll come back to this.)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 4
Ancient knots in art
While celtic knots began to appear in history around 450 AD...
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 5
Ancient knots in art
While celtic knots began to appear in history around 450 AD... ...knots have been appearing in art since at least 2200 BC.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 6
The story of the Gordian knot
A knot that was impossibly difficult to untie was tied to an
- xcart belonging to Gordias.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 7
The story of the Gordian knot
A knot that was impossibly difficult to untie was tied to an
- xcart belonging to Gordias.
An oracle proclaimed that the man who untied the knot would become king of Asia.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 8
The story of the Gordian knot
A knot that was impossibly difficult to untie was tied to an
- xcart belonging to Gordias.
An oracle proclaimed that the man who untied the knot would become king of Asia.
In 330 BC, Alexander the Great famously tried to untie the knot.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 9
The story of the Gordian knot
A knot that was impossibly difficult to untie was tied to an
- xcart belonging to Gordias.
An oracle proclaimed that the man who untied the knot would become king of Asia.
In 330 BC, Alexander the Great famously tried to untie the knot. Upon failing to solve the puzzle the “correct” way, he unsheathed his sword and sliced the knot in half!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 10
Knots and Incan accounting
The Inca empire in fourteenth century South America used knots (quipu) for accounting.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 11
Out to sea!
Knots have been put to use for fishing and sailing for as long as we can remember.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 12
Knots and chemistry
One of the first times knot theory appeared as a subject of scientific study was in 1860.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 13
Knots and chemistry
One of the first times knot theory appeared as a subject of scientific study was in 1860. Lord Kelvin, in an attempt to reconcile several competing atomic theories, proposed that atoms had a knotted structure.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 14
Knots and chemistry
One of the first times knot theory appeared as a subject of scientific study was in 1860. Lord Kelvin, in an attempt to reconcile several competing atomic theories, proposed that atoms had a knotted structure. Lord Kelvin and a scientist named Peter Tait set out to classify knots. This classification was meant to aid in the classification of atoms.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 15
Knots and chemistry (cont.)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 16
Recent history
More recently, knots have played a central role in the following disciplines.
1 Physics (quantum field theory, statistical mechanics) Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 17
Recent history
More recently, knots have played a central role in the following disciplines.
1 Physics (quantum field theory, statistical mechanics) 2 Chemistry (properties of molecules) Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 18
Recent history
More recently, knots have played a central role in the following disciplines.
1 Physics (quantum field theory, statistical mechanics) 2 Chemistry (properties of molecules) 3 Biology (DNA replication) Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 19
What is a knot?
Definition (A Mathematical Notion) A knot is a circle that doesn’t intersect itself sitting in space.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 20
What is a knot?
Definition (A Mathematical Notion) A knot is a circle that doesn’t intersect itself sitting in space. Intuitively, we say that two knots are equivalent if we can get from one to the other by bending, stretching, and rotating as long as we don’t break or cut the knot anywhere.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 21
What is a knot?
Definition (A Mathematical Notion) A knot is a circle that doesn’t intersect itself sitting in space. Intuitively, we say that two knots are equivalent if we can get from one to the other by bending, stretching, and rotating as long as we don’t break or cut the knot anywhere. (Sorry, Alexander. No swords allowed!)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 22
What are links?
Definition (A Mathematical Notion) A link is a collection of non-intersecting knots (perhaps linked with
- ne another) sitting in space.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 23
Trivialities
A trivial knot is called the unknot.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 24
Trivialities
A trivial knot is called the unknot. A trivial link is called the unlink.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 25
Knot diagrams
Because we like to represent knots using their pictures, we usually equate knots with their knot diagrams.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 26
Knot diagrams
Because we like to represent knots using their pictures, we usually equate knots with their knot diagrams. A knot diagram is a closed curve in the plane containing crossings (no tangencies or triple-points!). We decorate these crossings in a particular way to indicate which is the
- ver-strand and which is the under-strand of the crossing.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 27
Knot diagram equivalence
Problem: There are many different pictures of the same knot.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 28
Knot diagram equivalence
Problem: There are many different pictures of the same knot. For example, we can look at this knot...
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 29
Knot diagram equivalence
Problem: There are many different pictures of the same knot. For example, we can look at this knot... from our viewpoint ...
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 30
Knot diagram equivalence
Problem: There are many different pictures of the same knot. For example, we can look at this knot... from our viewpoint ... ...or from a “bird’s eye” viewpoint.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 31
Knot diagram equivalence
This knot...
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 32
Knot diagram equivalence
This knot... ...is the unknot in disguise!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 33
Knot diagram equivalence
How can we show that two diagrams represent the same knot?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 34
Reidemeister moves
In 1927, Kurt Reidemeister showed that knot diagrams are equivalent precisely when they can be related by the following moves.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 35
I’ll make an example of you!
Let’s use the example we looked at before to show how Reidemeister moves work.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 36
I’ll make an example of you!
Let’s use the example we looked at before to show how Reidemeister moves work.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 37
I’ll make an example of you!
Let’s use the example we looked at before to show how Reidemeister moves work.
At the end of our sequence of moves, we have the mirror image of the diagram we wanted.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 38
I’ll make an example of you!
Let’s use the example we looked at before to show how Reidemeister moves work.
At the end of our sequence of moves, we have the mirror image of the diagram we wanted. This knot, called the figure eight knot, is equivalent to it’s mirror image. (Prove it!)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 39
A slippery slope
Just as the same knot can look very different in two different diagrams, different knots can look very similar to one another.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 40
A slippery slope
Just as the same knot can look very different in two different diagrams, different knots can look very similar to one another. Unknotted Knotted
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 41
A slippery slope
Just as the same knot can look very different in two different diagrams, different knots can look very similar to one another. Unknotted? Knotted?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 42
A slippery slope
Just as the same knot can look very different in two different diagrams, different knots can look very similar to one another. Unknotted?? Knotted??
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 43
Let’s use this idea to play a game!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 44
Knotting vs. Unknotting
Starting with a knot that is missing its crossing information, we can play the Knotting–Unknotting Game.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 45
Knotting vs. Unknotting
Starting with a knot that is missing its crossing information, we can play the Knotting–Unknotting Game. In this game, there are two players, Knot and Unknot.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 46
Knotting vs. Unknotting
Starting with a knot that is missing its crossing information, we can play the Knotting–Unknotting Game. In this game, there are two players, Knot and Unknot. Players take turns choosing crossing information.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 47
Knotting vs. Unknotting
Starting with a knot that is missing its crossing information, we can play the Knotting–Unknotting Game. In this game, there are two players, Knot and Unknot. Players take turns choosing crossing information. Knot wants to make something that is knotted up, while Unknot wants to make something that can be untangled.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 48
Playing the game
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 49
Unknot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 50
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 51
Unknot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 52
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 53
Unknot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 54
Unknot’s Move Who wins?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 55
Now it’s your turn
1 Get a worksheet and find a partner. Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 56
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 57
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 58
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. 4 Play again on the same “game board,” switching who goes
first but keeping the same roles.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 59
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. 4 Play again on the same “game board,” switching who goes
first but keeping the same roles.
5 When you are done, draw your own game board and play
another game!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 60
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. 4 Play again on the same “game board,” switching who goes
first but keeping the same roles.
5 When you are done, draw your own game board and play
another game!
6 Did you learn any strategies? Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 61
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. 4 Play again on the same “game board,” switching who goes
first but keeping the same roles.
5 When you are done, draw your own game board and play
another game!
6 Did you learn any strategies? 7 Any observations about which player has an advantage? Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 62
Now it’s your turn
1 Get a worksheet and find a partner. 2 Assign roles.
(One of you is Knot and one of you is Unknot.)
3 Decide who plays first, then play your first game. 4 Play again on the same “game board,” switching who goes
first but keeping the same roles.
5 When you are done, draw your own game board and play
another game!
6 Did you learn any strategies? 7 Any observations about which player has an advantage? 8 Did any questions arise? Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 63
Some observations
If Knot can make the knot alternating, she can win.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 64
Some observations
If Knot can make the knot alternating, she can win. If Unknot can make long strands go entirely under or entirely
- ver, the knot can be simplified.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 65
A winning strategy
If both players play optimally on this game board, whoever goes first loses. This is true regardless of whether Knot goes first or Unknot goes first!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 66
A winning strategy
If both players play optimally on this game board, whoever goes first loses. This is true regardless of whether Knot goes first or Unknot goes first! What about your game board? Who has a winning strategy?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 67
What other games could we play?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 68
Knotting vs. Linking
If you start with a knot or a link that is missing its crossing information, you can play the Link Smoothing Game.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 69
Knotting vs. Linking
If you start with a knot or a link that is missing its crossing information, you can play the Link Smoothing Game. In this game, there are two players: Knot and Link.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 70
Knotting vs. Linking
If you start with a knot or a link that is missing its crossing information, you can play the Link Smoothing Game. In this game, there are two players: Knot and Link. Players take turns to select a crossing and smooth it:
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 71
Knotting vs. Linking
If you start with a knot or a link that is missing its crossing information, you can play the Link Smoothing Game. In this game, there are two players: Knot and Link. Players take turns to select a crossing and smooth it: Link wants to disconnect the the diagram to get an unlink, while Knot wants to keep it all in one piece to get an unknot.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 72
Playing the game
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 73
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 74
Link’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 75
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 76
Link’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 77
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 78
Link’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 79
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 80
Link’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 81
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 82
Link’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 83
Knot’s Move
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 84
Knot’s Move Who wins?
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 85
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 86
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
2 Decide who plays first, and play the game. Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 87
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
2 Decide who plays first, and play the game. 3 Play again, switching who goes first. Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 88
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
2 Decide who plays first, and play the game. 3 Play again, switching who goes first. 4 When you are done, draw your own game board and play
another game!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 89
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
2 Decide who plays first, and play the game. 3 Play again, switching who goes first. 4 When you are done, draw your own game board and play
another game!
5 Any observations about which player has an advantage? Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 90
Now it’s your turn
1 Assign roles.
(One of you is Knot and one of you is Link.)
2 Decide who plays first, and play the game. 3 Play again, switching who goes first. 4 When you are done, draw your own game board and play
another game!
5 Any observations about which player has an advantage? 6 Did any questions arise? Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 91
Some observations
Link wins if he can play on a nugatory crossing.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 92
Some observations
Link wins if he can play on a nugatory crossing. Link wins if the diagram contains a picture like this:
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 93
Some observations
Link wins if he can play on a nugatory crossing. Link wins if the diagram contains a picture like this: When Link plays last, Link wins.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 94
Some observations
Link wins if he can play on a nugatory crossing. Link wins if the diagram contains a picture like this: When Link plays last, Link wins. Does Link always have the upper hand??
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 95
A winning strategy
This is an example of a link shadow where Knot actually has a winning strategy if she plays second.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 96
A winning strategy
This is an example of a link shadow where Knot actually has a winning strategy if she plays second. More often than not, Link has a winning strategy...
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 97
A winning strategy
This is an example of a link shadow where Knot actually has a winning strategy if she plays second. More often than not, Link has a winning strategy...but we have found infinite families of diagrams on which Knot has an advantage.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 98
Unsolicited advice
Keep playing these games and see if you can figure out who has a winning strategy for specific shadows!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 99
Unsolicited advice
Keep playing these games and see if you can figure out who has a winning strategy for specific shadows! These are just a couple of games you could play using knots and links. Invent your own games!
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 100
Unsolicited advice
Keep playing these games and see if you can figure out who has a winning strategy for specific shadows! These are just a couple of games you could play using knots and links. Invent your own games! Are you interested in knowing more about knots? Read Why Knot?, a comic book about knots by Colin Adams.
Allison Henrich, Ph.D. A Tale of Knots & Games
SLIDE 101
thank you. thank you. thank you.
Allison Henrich, Ph.D. A Tale of Knots & Games