The Human Knot

1. The Human Knot Capstone-Mathematics California Lutheran University Spring 2005 Lucas Lembrick With: Dr. Cynthia Wyels and Maggy Tomova

2. How to Play the Human Knot Game • A group of people stand in a circle facing in. • Everybody puts both hands in and grabs two more hands making sure not to grab the hands of somebody standing next to them. • They now form a knot. • The goal is to untangle themselves until they have the unknot.

3. Question: • Does a simple set of directions exist that when given to a group of people they will always be able to form the unknot while playing the human knot game?

4. Definitions Knot: A knot is a closed, one dimensional, and non-intersecting curve in three dimensional space. Unknot: The unknot, also known as the "trivial knot", is simply a circle embedded in three-dimensional space with no crossings. Link: A link is a group of knots or unknots embedded in three dimensional space. Each knot or unknot embedded in the link is called a component.

5. The Beginning • It is obvious that while playing this game that the unknot is not always going to be the result. Sometimes you will get other knots and sometimes even links. • At first we looked at other questions relating to our underlying goal such as: • Is it possible to ensure a link?

6. Answer: • Yes. • Quite simply, number the participants and tell them that if they are an even number they can only hold hands with other even numbers and if they are an odd number they can only hold hands with other odds. • This will ensure that you have a two component link, one component made up of even people and one made up of odd people.

7. Next we asked: Is it possible to guarantee a knot and not a link? • This is when we changed the rules of the game. • We decided to start with three people forming an unknot and then breaking their hands at one location and adding another person, thereby always forming a knot and not a link.

8. How we did this • We started with our three people crossing hands so they formed the attached picture where the lines are their bodies and the dots are their hands.

9. We then broke one of the hand holdings apart and added a fourth person in one of the six following ways.

10. Definitions • Reidermeister Moves (RI, RII, RIII): • RI: • RII: • RIII:

11. Using Reidermeister moves we will attempt to untangle these knots Unknot Trefoil

12. OR • In fact, we discovered that all six pictures are isomorphic to: Unknot Trefoil

13. Definition • Oriented: An oriented knot or link is any knot or link that has direction. The direction is referred to as the orientation and is denoted by arrows in a knot diagram. • Crossing Sign: At a crossing of an oriented knot, take the part that is an under crossing and rotate it clockwise until it is lined up with the over crossing. If the two orientations match then the crossing is said to be positive (+), if they do not match the crossing is negative (-). - +

14. Question: What is the difference between the first and second picture? • Moving away from the Human Knot…. if we orient the knot and give the original crossing a positive or negative value based on the orientation, then we make the new crossings in such a way that they have signs opposite to that of the original crossing, then we will have the unknot.

15. Unknot Trefoil + ? - ? ? + - ? - - The first picture is now the unknot and the second is the trefoil. Add fourth person according to crossing sign. Figure out crossing sign for new crossings. Calculate crossing sign. Orient the knots.

16. Definitions • Arc of a knot: The part of a knot in between two crossings. • Adjacent Crossings: Two crossings connected by an arc.

17. 3 2 1 Algorithm to create unknot 1. If we number the original crossing (1) and then as we make the two new crossings numbering then (2) and (3) respectively we will have a oriented knot with numbered crossings in the order that we made them.

18. 2. Next cut your knot in any spot and thread a new piece into the knot with the following instructions: 3. When you are making a new crossing, observe the two adjacent crossings and look at the crossing sign of the lowest numbered crossing. 4. Make your new crossing the opposite sign, and number it with the next consecutive number. 5. Do this until you are ready to reconnect the knot. 6. Repeat steps 2-6

20. Problems • This algorithm works most of the time, but for it to work all the time it will need a few modifications. When working on a proof of it I found some counterexamples to the algorithm that I believe could easily be taken care of, but then the simplicity of it is lost.

21. Examples of Problem Knots

23. Definitions • Stick knot: A knot formed out of straight sticks, each one connected to another at a vertex.

24. New Direction • After meeting with Colin Adams he suggested that we look at different stick knots where the vertices of the sticks formed a circle and no two vertices adjacent on the circle were connected with a stick.

25. B C A D E Human Stick Knots • Human Stick Knot: A stick knot inscribed in a circle such that all vertices intersect the circle and no two consecutive vertices are adjacent. • Consecutive vertices: Two vertices on the circle where there does not exist a vertex between the two on the circle. • Adjacent vertices: Two vertices connected by a stick.

26. More Definitions • Stick Number (s(K)): The minimum number of sticks necessary to make a specific stick knot (K). • Human Stick Number (Hs(K)): The minimum number of sticks necessary to make a specific human stick knot (K).

27. Lemma: Instances of consecutive adjacent vertices may be reduced exactly when there exists a vertex adjacent to neither vertex in a consecutive adjacent pair. • Proof: Let (v,u) be a pair of consecutive adjacent vertices and w be a vertex adjacent to neither v nor u. Then w can be places on the circle between v and u by a series of Reidermeister moves to eliminate the consecutiveness of v and u. • If no such w exists, then without loss of generality let z be adjacent to u. By placing z in between u and v, the consecutive adjacent pair (v,z) will be created.

28. B C A D E

29. Theorem: Hs(unknot)=5 • Proof: • s(unknot)=3, this is a lower bound for Hs(unkot). • Inscribe the three-stick unknot in a circle. Each pair of adjacent vertices will be consecutive. • Inscribe the four-stick unknot in a circle. At most two pairs of adjacent vertices will be non-consecutive, leaving two pairs of consecutive adjacent vertices.

30. The Five-Stick Human Unknot

31. Question: How does Hs(k) differ from s(k) in other knots? • To answer this I looked at multiple other stick knots and, using Reidermeister moves, put them in a human knot projection. • With the exception of the unknot, all other knots I examined satisfied Hs(k)=s(k).

33. Conjecture: Hs(k)=s(k) for k unknot • Ideas for proof: • Since the smallest s(k) for any knot other than the unknot is six (trefoil), for any two adjacent vertices, u and v, there will always exist a vertex w not adjacent to either of them. • What we were mainly working on to prove this conjecture was to place a general stick knot inside of a circle and showing that you can “pull” each vertex out to intersect the circle without disrupting the stick number.

34. Further Questions for Investigation • Are there simple changes that can be made to the algorithm to make it work? • Is s(K) in fact equal to Hs(K) for all knots other than the unknot? • There are 35 knots that can be formed with 10 sticks or less, if you consider a person to be five sticks, how many of these knots can actually be made with two people? • Does a simple set of rules exist to guarantee the unknot in the human knot game?

35. Adams, Colin. The Knot Book. N.p.: American Mathematical Society, 2004. • ‑ ‑ ‑. Personal interview. 7 Mar. 2005. • Foisey, Joel. “human knot.” E‑mail to Lucas Lembrick. 21 Oct. 2004. • “Glossary of Terms.” Knot Theory Home Page. Thinkquest. 12 Apr. 2005 <http://library.thinkquest.org/12295/main.html>. • Payne, Bryson R. “Advanced Knot Theory Topics.” Knot Theory: The Website for learning more about knots. North Georgia College and State University. 21 Apr. 2005 <http://www.freelearning.com/knots/advanced.htm>. • Rawdon, Eric J. “Equalateral Stick Number.” Knot Theory with Knot Plot. 11 Feb. 2004. Center for Experimental and Constructive Mathematics. 12 Mar. 2005 <http://www.colab.sfu.ca/KnotPlot/ktheory.html>. • “Table of Knots.” Pop Math. 2002. Mathematics and Knots, U.C.N.W., Bangor. 21 Apr. 2005 <http://www.popmath.org.uk/exhib/pagesexhib/table.ht>.

36. Special Thanks • Dr. Colin Adams-Williams College • Dr. Karrolyne Fogel-California Lutheran University • Dr. Joel Foisy-SUNY Potsdam • Maggy Tomova-UC Santa Barbara • Dr. Cynthia Wyels-California Lutheran University