Proving the Trefoil is Knotted
William Kuszmaul, Rohil Prasad, Isaac Xia
Mentored by Umut Varolgunes
Fourth Annual MIT PRIMES Conference
May 17, 2014
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H OW T O B UILD A K NOT
1. Take 1 piece of string
2. Tangle it up
3. Glue ends together
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E XAMPLE K NOTS
(a) Trefoil
(b) 41
(c) 106
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S OME KNOTS ARE THE SAME
Knot diagrams can be deformed, like in real life.
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R EIDEMEISTER M OVES
I
We think of knots as knot diagrams:
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T HE BIG QUESTION
Question: How can we show that two knots aren’t equal?
Answer: Find an Invariant.
Assign a number to each knot diagram so that two knot
diagrams that are equivalent have same assigned number.
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E XAMPLE I NVARIANT: C ROSSING N UMBER
I
Find an equivalent knot diagram with the fewest crossings
I
Crossing number = fewest number of crossings
I
But this is hard to compute in reality
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A B ETTER I NVARIANT: T RICOLORING K NOTS
I
Assign one of three colors, a, b, c to each strand in a knot
diagram (red, blue, green)
I
At any crossing, strands must either have all different or
all same color
I
We are interested in the number of tricolorings of a knot.
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W HY T RICOLORABILITY M ATTERS
Theorem
If a knot diagram has k tricolorings, then all equivalent knot diagrams
have k tricolorings.
How to Prove:
Show number of tricolorings is maintained by Reidemeister
moves.
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E XAMPLE : B IJECTING T RICOLORINGS F OR S ECOND
R EIDEMEISTER M OVE
Case 1: Same Color
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E XAMPLE : B IJECTING T RICOLORINGS F OR S ECOND
R EIDEMEISTER M OVE
Case 2: Different Colors
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T REFOIL IS K NOTTED !
(a) 3 Unknot
Tricolorings
(b) 9 Trefoil
Tricolorings
=⇒ Trefoil6=Unknot
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A D IFFERENT A PPROACH : F OCUS ON O NE C ROSSING
Trefoil Knot
(a) Crossing Change 1
(b) Crossing Change 2
How can we take advantage of this?
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A W EIRD P OLYNOMIAL : J ONES P OLYNOMIAL J(K)
1. Pick a crossing:
2. Look at its rearrangements:
(a) K1
(b) K2
(c) K3
3. Use recursion:
t−2 J(K1 ) − t2 J(K2 ) = (t − t−1 )J(K3 )
J(O) = 1
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W HY J ONES P OLYNOMIAL M ATTERS : I T ’ S AN
I NVARIANT !
Theorem
If knot diagrams A and B are equivalent, then J(A) = J(B).
How to Prove: Show Jones Polynomial is unchanged by
Reidemeister Moves.
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E XAMPLE : 2 U NKNOTS AT O NCE
(a) A1
(b) A2
(c) A3 (2 Unknots)
J(2 Unknots) = J(A3 )
t−2 J(A1 ) − t2 J(A2 )
t − t−1
−2
t J(Unknot) − t2 J(Unknot)
=
t − t−1
−2
2
t (1) − t (1)
=
t − t−1
=
= −t − t−1 .
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A W EIRDER E XAMPLE : L INKED U NKNOTS
(a) (Linked Unknots) B1
(b) B2
(c) B3
J(Linked Unknots) = J(B1 )
= t4 J(B2 ) + (t3 − t)J(B3 )
= t4 J(Separate Unknots) + (t3 − t)J(Unknot)
= t4 (−t − t−1 ) + (t3 − t)(1)
= −t5 − t.
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J ONES P OLYNOMIAL OF T REFOIL
(a) T1 (Trefoil)
(b) T2
(c) T3
J(Trefoil) = J(T1 )
= t4 J(T2 ) + (t3 − t)J(T3 )
= t4 J(Unknot) + (t3 − t)J(Linked Unknots)
= t4 (1) + (t3 − t)(−t5 − t)
= −t8 + t6 + t2 .
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C OMPLETING S ECOND P ROOF
=⇒ Trefoil6=Unknot
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A CKNOWLEDGEMENTS
We want to thank
1. Our mentor Umut Varolgunes for working with us every
week.
2. MIT PRIMES for setting up such a fun reading group.
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