The Human The Human KnotKnot

Lucas LembrickLucas Lembrick

With: Dr. Cynthia Wyels and With: Dr. Cynthia Wyels and Maggy TomovaMaggy Tomova

Capstone-MathematicsCalifornia Lutheran University

Spring 2005

How to Play the Human How to Play the Human Knot GameKnot Game

• A group of people stand in a circle facing in.A group of people stand in a circle facing in.

• Everybody puts both hands in and grabs two Everybody puts both hands in and grabs two more hands making sure not to grab the more hands making sure not to grab the hands of somebody standing next to them.hands of somebody standing next to them.

• They now form a knot. They now form a knot.

• The goal is to untangle themselves until they The goal is to untangle themselves until they have the unknot.have the unknot.

Question:Question:

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

DefinitionsDefinitions

Knot: A Knot: A knotknot is a closed, one is a closed, one dimensional, and non-intersecting dimensional, and non-intersecting curve in three dimensional space. curve in three dimensional space.

Unknot: The Unknot: The unknotunknot, also known , also known as the "trivial knot", is simply a as the "trivial knot", is simply a circle embedded in three-circle embedded in three-dimensional space with no dimensional space with no crossings.crossings.

Link: A Link: A linklink is a group of knots or is a group of knots or unknots embedded in three unknots embedded in three dimensional space. Each knot or dimensional space. Each knot or unknot embedded in the unknot embedded in the linklink is is called a component. called a component.

The BeginningThe Beginning

• It is obvious that while playing this game It is obvious that while playing this game that the unknot is not always going to be that the unknot is not always going to be the result. Sometimes you will get other the result. Sometimes you will get other knots and sometimes even links.knots and sometimes even links.

• At first we looked at other questions At first we looked at other questions relating to our underlying goal such as:relating to our underlying goal such as:

–Is it possible to ensure a Is it possible to ensure a link?link?

Answer:Answer:

• Yes.Yes.• Quite simply, number the participants Quite simply, number the participants

and tell them that if they are an even and tell them that if they are an even number they can only hold hands with number they can only hold hands with other even numbers and if they are an other even numbers and if they are an odd number they can only hold hands odd number they can only hold hands with other odds.with other odds.

• This will ensure that you have a two This will ensure that you have a two component link, one component made up component link, one component made up of even people and one made up of odd of even people and one made up of odd people.people.

• Next we asked: Is it possible to Next we asked: Is it possible to guarantee a knot and not a link?guarantee a knot and not a link?

• This is when we changed the rules of This is when we changed the rules of the game.the game.

• We decided to start with three We decided to start with three people forming an unknot and then people forming an unknot and then breaking their hands at one location breaking their hands at one location and adding another person, thereby and adding another person, thereby always forming a knot and not a link.always forming a knot and not a link.

How we did thisHow we did this

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

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

DefinitionsDefinitions

• Reidermeister Moves (RI, RII, RIII):Reidermeister Moves (RI, RII, RIII):– RI:RI:– – – RII:RII:– – – RIII:RIII:

–

Unknot Trefoil

Using Reidermeister moves we will attempt to untangle these knots

• In fact, we discovered that all six pictures In fact, we discovered that all six pictures are isomorphic to:are isomorphic to:

Unknot Trefoil

OR

DefinitionDefinition

• Oriented: An Oriented: An orientedoriented knot or knot or link is any knot or link that has link is any knot or link that has direction. The direction is direction. The direction is referred to as the orientation and referred to as the orientation and is denoted by arrows in a knot is denoted by arrows in a knot diagram. diagram.

• Crossing Sign: At a crossing of Crossing Sign: At a crossing of an oriented knot, take the part an oriented knot, take the part that is an under crossing and that is an under crossing and rotate it clockwise until it is lined rotate it clockwise until it is lined up with the over crossing. If the up with the over crossing. If the two orientations match then the two orientations match then the crossing is said to be positive (+), crossing is said to be positive (+), if they do not match the crossing if they do not match the crossing is negative (-).is negative (-).

+

-

Question: What is the Question: What is the difference between the first difference between the first

and second picture?and second picture?

• Moving away from the Human Moving away from the Human Knot…. if we orient the knot and Knot…. if we orient the knot and give the original crossing a positive give the original crossing a positive or negative value based on the or negative value based on the orientation, then we make the new orientation, then we make the new crossings in such a way that they crossings in such a way that they have signs opposite to that of the have signs opposite to that of the original crossing, then we will have original crossing, then we will have the unknot.the unknot.

? ? ? ?

- -

Orient the knots.Calculate crossing sign.Figure out crossing sign for new crossings.

++

- -

Add fourth person according to crossing sign.The first picture is now the unknot and the second is the trefoil.

Unknot Trefoil

DefinitionsDefinitions

• Arc of a knot: The part of a knot in Arc of a knot: The part of a knot in between two crossings.between two crossings.

• Adjacent Crossings: Two crossings Adjacent Crossings: Two crossings connected by an arc.connected by an arc.

Algorithm to create Algorithm to create unknotunknot

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

1

23

2.2. Next cut your knot in any spot and Next cut your knot in any spot and thread a new thread a new piece into the knot with the piece into the knot with the following following instructions:instructions:

3.3. When you are making a new crossing, When you are making a new crossing, observe the two adjacent crossings observe the two adjacent crossings

and look and look at at the crossing sign of the the crossing sign of the lowest numbered lowest numbered crossing. crossing.

4.4. Make your new crossing the opposite Make your new crossing the opposite sign, and sign, and number it with the next number it with the next consecutive number.consecutive number.

5.5. Do this until you are ready to Do this until you are ready to reconnect the reconnect the knot.knot.

6.6. Repeat steps 2-6Repeat steps 2-6

ProblemsProblems

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

Examples of Problem Examples of Problem KnotsKnots

DefinitionsDefinitions

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

New DirectionNew Direction

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

Human Stick KnotsHuman Stick Knots• Human Stick Knot: A stick knot inscribed in Human Stick Knot: A stick knot inscribed in

a circle such that all vertices intersect the a circle such that all vertices intersect the circle and no two consecutive vertices are circle and no two consecutive vertices are adjacent.adjacent.

• Consecutive vertices: Two vertices on the Consecutive vertices: Two vertices on the circle where there does not exist a vertex circle where there does not exist a vertex between the two on the circle.between the two on the circle.

• Adjacent vertices: Two vertices connected Adjacent vertices: Two vertices connected by a stick.by a stick.

A

B

C

D

E

More DefinitionsMore Definitions

• Stick Number (sStick Number (s(K)):(K)): The minimum The minimum number of sticks necessary to make number of sticks necessary to make a specific stick knot a specific stick knot (K).(K).

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

Lemma: Instances of consecutive Lemma: Instances of consecutive adjacent vertices may be reduced adjacent vertices may be reduced exactly when there exists a vertex exactly when there exists a vertex

adjacent to neither vertex in a adjacent to neither vertex in a consecutive adjacent pair.consecutive adjacent pair.

• Proof: Let (Proof: Let (v,uv,u) be a pair of consecutive ) be a pair of consecutive adjacent vertices and adjacent vertices and ww be a vertex adjacent to be a vertex adjacent to neither neither vv nor nor uu. Then . Then ww can be places on the can be places on the circle between circle between v v and and uu by a series of by a series of Reidermeister moves to eliminate the Reidermeister moves to eliminate the consecutiveness of consecutiveness of vv and and uu..

• If no such If no such ww exists, then without loss of exists, then without loss of generality let generality let zz be adjacent to be adjacent to uu. By placing . By placing zz in between in between uu and and vv, the consecutive adjacent , the consecutive adjacent pair (pair (v,zv,z) will be created.) will be created.

A

B

C

D

E

Theorem: Hs(unknot)=5Theorem: Hs(unknot)=5

• Proof: Proof: • s(unknot)=3, this is a lower bound for s(unknot)=3, this is a lower bound for

Hs(unkot). Hs(unkot). • Inscribe the three-stick unknot in a Inscribe the three-stick unknot in a

circle. Each pair of adjacent vertices circle. Each pair of adjacent vertices will be consecutive.will be consecutive.

• Inscribe the four-stick unknot in a circle. Inscribe the four-stick unknot in a circle. At most two pairs of adjacent vertices At most two pairs of adjacent vertices will be non-consecutive, leaving two will be non-consecutive, leaving two pairs of consecutive adjacent vertices. pairs of consecutive adjacent vertices.

The Five-Stick Human The Five-Stick Human UnknotUnknot

Question: How does Hs(k) Question: How does Hs(k) differ from s(k) in other differ from s(k) in other

knots?knots?• To answer this I looked at multiple To answer this I looked at multiple

other stick knots and, using other stick knots and, using Reidermeister moves, put them in a Reidermeister moves, put them in a human knot projection. human knot projection.

• With the exception of the unknot, all With the exception of the unknot, all other knots I examined satisfied other knots I examined satisfied Hs(k)=s(k).Hs(k)=s(k).

Conjecture: Hs(Conjecture: Hs(kk)=s(k) for )=s(k) for k unknotk unknot

• Ideas for proof:Ideas for proof:– Since the smallest s(k) for any knot other Since the smallest s(k) for any knot other

than the unknot is six (trefoil), for any two than the unknot is six (trefoil), for any two adjacent vertices, u and v, there will always adjacent vertices, u and v, there will always exist a vertex w not adjacent to either of exist a vertex w not adjacent to either of them.them.

– What we were mainly working on to prove What we were mainly working on to prove this conjecture was to place a general stick this conjecture was to place a general stick knot inside of a circle and showing that you knot inside of a circle and showing that you can “pull” each vertex out to intersect the can “pull” each vertex out to intersect the circle without disrupting the stick number.circle without disrupting the stick number.

Further Questions for Further Questions for InvestigationInvestigation

• Are there simple changes that can be made to the Are there simple changes that can be made to the algorithm to make it work?algorithm to make it work?

• Is s(K) in fact equal to Hs(K) for all knots other than Is s(K) in fact equal to Hs(K) for all knots other than the unknot?the unknot?

• There are 35 knots that can be formed with 10 sticks There are 35 knots that can be formed with 10 sticks or less, if you consider a person to be five sticks, or less, if you consider a person to be five sticks, how many of these knots can actually be made with how many of these knots can actually be made with two people?two people?

• Does a simple set of rules exist to guarantee the Does a simple set of rules exist to guarantee the unknot in the human knot game?unknot in the human knot game?

• Adams, Colin. Adams, Colin. The Knot BookThe Knot Book. N.p.: American . N.p.: American Mathematical Society, 2004.Mathematical Society, 2004.

• ‑ ‑ ‑‑ ‑ ‑. Personal interview. 7 Mar. 2005.. Personal interview. 7 Mar. 2005.• Foisey, Joel. “human knot.” E‑mail to Lucas Foisey, Joel. “human knot.” E‑mail to Lucas

Lembrick. 21 Lembrick. 21 Oct. Oct. 2004.2004.• ““Glossary of Terms.” Glossary of Terms.” Knot Theory Home PageKnot Theory Home Page. .

Thinkquest. Thinkquest. 12 Apr. 2005 12 Apr. 2005 <http://library.thinkquest.org/12295/main.html>.<http://library.thinkquest.org/12295/main.html>.

• Payne, Bryson R. “Advanced Knot Theory Topics.” Payne, Bryson R. “Advanced Knot Theory Topics.” Knot Knot Theory: The Website for learning more about Theory: The Website for learning more about knotsknots. . North Georgia College and State North Georgia College and State University. 21 Apr. University. 21 Apr. 2005 2005 <http://www.freelearning.com/knots/advanced.htm><http://www.freelearning.com/knots/advanced.htm>..

• Rawdon, Eric J. “Equalateral Stick Number.” Rawdon, Eric J. “Equalateral Stick Number.” Knot Knot Theory Theory with Knot Plotwith Knot Plot. 11 Feb. 2004. Center for . 11 Feb. 2004. Center for Experimental Experimental and Constructive Mathematics. 12 and Constructive Mathematics. 12 Mar. 2005 Mar. 2005 <http://www.colab.sfu.ca/KnotPlot/ktheory.html>.<http://www.colab.sfu.ca/KnotPlot/ktheory.html>.

• ““Table of Knots.” Table of Knots.” Pop MathPop Math. 2002. Mathematics and . 2002. Mathematics and Knots, Knots, U.C.N.W., Bangor. 21 Apr. 2005 U.C.N.W., Bangor. 21 Apr. 2005 <http://www.popmath.org.uk/exhib/pagesexhib/table<http://www.popmath.org.uk/exhib/pagesexhib/table.ht>..ht>.

Special ThanksSpecial Thanks

• Dr. Colin Adams-Williams College Dr. Colin Adams-Williams College

• Dr. Karrolyne Fogel-California Lutheran Dr. Karrolyne Fogel-California Lutheran UniversityUniversity

• Dr. Joel Foisy-SUNY PotsdamDr. Joel Foisy-SUNY Potsdam

• Maggy Tomova-UC Santa BarbaraMaggy Tomova-UC Santa Barbara

• Dr. Cynthia Wyels-California Lutheran Dr. Cynthia Wyels-California Lutheran UniversityUniversity