Untangling the Unknot

Colby College Digital Commons @ Colby

CLAS: Colby Liberal Arts Symposium

Apr 30th, 3:00 PM - 3:25 PM

Untangling the Unknot

Amar Sehic Colby College

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Sehic, Amar, "Untangling the Unknot" (2015). CLAS: Colby Liberal Arts Symposium. 348. https://digitalcommons.colby.edu/clas/2015/program/348

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Untangling the Unknot

Amar Sehic

April 29, 2015 We make life simpler by thinking in 2D.

Untangling the Unknot

What are knots?

Intuitively: Take a piece of string, tangle it up, glue the ends. Untangling the Unknot

What are knots?

Intuitively: Take a piece of string, tangle it up, glue the ends. We make life simpler by thinking in 2D. Theorem (Reidemeister 1926) Let K and L be two knot diagrams of the same knot. Then, there exists a sequence of Reidemeister moves relating K to L.

Untangling the Unknot

Reidemeister moves

We capture the idea of changing our knot with the 3 types of Reidemeister moves. Theorem (Reidemeister 1926) Let K and L be two knot diagrams of the same knot. Then, there exists a sequence of Reidemeister moves relating K to L.

Untangling the Unknot

Reidemeister moves

We capture the idea of changing our knot with the 3 types of Reidemeister moves. Theorem (Reidemeister 1926) Let K and L be two knot diagrams of the same knot. Then, there exists a sequence of Reidemeister moves relating K to L.

Untangling the Unknot

Reidemeister moves

We capture the idea of changing our knot with the 3 types of Reidemeister moves. Theorem (Reidemeister 1926) Let K and L be two knot diagrams of the same knot. Then, there exists a sequence of Reidemeister moves relating K to L.

Untangling the Unknot

Reidemeister moves

We capture the idea of changing our knot with the 3 types of Reidemeister moves. Untangling the Unknot

Reidemeister moves

We capture the idea of changing our knot with the 3 types of Reidemeister moves.

Theorem (Reidemeister 1926) Let K and L be two knot diagrams of the same knot. Then, there exists a sequence of Reidemeister moves relating K to L. How about some fancy classification? Two neat invariants: Crossing Number and Bridge Number

Untangling the Unknot

Telling Knots Apart

Q: When are two knot diagrams the same? How about some fancy classification? Two neat invariants: Crossing Number and Bridge Number

Untangling the Unknot

Telling Knots Apart

Q: When are two knot diagrams the same? Two neat invariants: Crossing Number and Bridge Number

Untangling the Unknot

Telling Knots Apart

Q: When are two knot diagrams the same?

How about some fancy classification? Untangling the Unknot

Telling Knots Apart

Q: When are two knot diagrams the same?

How about some fancy classification? Two neat invariants: Crossing Number and Bridge Number Bridge Number b(K)

Figure: A diagram with cr(K)=4, b(K)=2

Untangling the Unknot

Knot Invariants

Crossing Number cr(K) Untangling the Unknot

Knot Invariants

Crossing Number cr(K) Bridge Number b(K)

Figure: A diagram with cr(K)=4, b(K)=2 Simpler Q: When is a knot diagram actually a diagram of the unknot?

Figure: The Unknot

Untangling the Unknot

Asking A Simpler Question

Q: When are two knot diagrams the same? Figure: The Unknot

Untangling the Unknot

Asking A Simpler Question

Q: When are two knot diagrams the same? Simpler Q: When is a knot diagram actually a diagram of the unknot? Untangling the Unknot

Asking A Simpler Question

Q: When are two knot diagrams the same? Simpler Q: When is a knot diagram actually a diagram of the unknot?

Figure: The Unknot A: Yes!

Untangling the Unknot

Knot or Not?

Q: Is this a diagram of the unknot? Untangling the Unknot

Knot or Not?

Q: Is this a diagram of the unknot?

A: Yes! A: Yes! But very hard to unknot!

Untangling the Unknot

Knot or Not?

Q: Is this a diagram of the unknot? Untangling the Unknot

Knot or Not?

Q: Is this a diagram of the unknot?

A: Yes! But very hard to unknot! Q: How many Reidemeister moves? Theorem (Henrich-Kauffman) Let K be a diagram of the unknot. Let M = 2b(K) + 2cr(K). Then, the number of Reidemeister moves needed to unknot K is less than or equal to

M X 1 i[(i − 1)!]2(M − 2) 2 i=2

Untangling the Unknot

The Unknotting Problem

This is a very hard problem! Theorem (Henrich-Kauffman) Let K be a diagram of the unknot. Let M = 2b(K) + 2cr(K). Then, the number of Reidemeister moves needed to unknot K is less than or equal to

M X 1 i[(i − 1)!]2(M − 2) 2 i=2

Untangling the Unknot

The Unknotting Problem

This is a very hard problem! Q: How many Reidemeister moves? Untangling the Unknot

The Unknotting Problem

This is a very hard problem! Q: How many Reidemeister moves? Theorem (Henrich-Kauffman) Let K be a diagram of the unknot. Let M = 2b(K) + 2cr(K). Then, the number of Reidemeister moves needed to unknot K is less than or equal to

M X 1 i[(i − 1)!]2(M − 2) 2 i=2 Yes! Theorem (Lackenby) Let K be a diagram of the unknot with c crossings. Then there is a sequence of at most (236c)11 Reidemeister moves that transforms K into the trivial diagram.

How was this done?

Untangling the Unknot

The Unknotting Problem

Can we do better? Theorem (Lackenby) Let K be a diagram of the unknot with c crossings. Then there is a sequence of at most (236c)11 Reidemeister moves that transforms K into the trivial diagram.

How was this done?

Untangling the Unknot

The Unknotting Problem

Can we do better? Yes! How was this done?

Untangling the Unknot

The Unknotting Problem

Can we do better? Yes! Theorem (Lackenby) Let K be a diagram of the unknot with c crossings. Then there is a sequence of at most (236c)11 Reidemeister moves that transforms K into the trivial diagram. Untangling the Unknot

The Unknotting Problem

Can we do better? Yes! Theorem (Lackenby) Let K be a diagram of the unknot with c crossings. Then there is a sequence of at most (236c)11 Reidemeister moves that transforms K into the trivial diagram.

How was this done? Paying attention to crossings:

Untangling the Unknot

Arc Presentations

We simplify even more! Paying attention to crossings:

Untangling the Unknot

Arc Presentations

We simplify even more! Untangling the Unknot

Arc Presentations

We simplify even more!

Paying attention to crossings: Untangling the Unknot

Arc Presentations

An example of fixing our arc-presentation: We also have two ways to relate arc presentations of the same knot. Stabilization/Destabilization Moves:

Untangling the Unknot

New Invariant, New Moves

We let K be an arc presentation. Then cl(K) is the number of vertical arcs in the diagram. Untangling the Unknot

New Invariant, New Moves

We let K be an arc presentation. Then cl(K) is the number of vertical arcs in the diagram. We also have two ways to relate arc presentations of the same knot. Stabilization/Destabilization Moves: Untangling the Unknot

New Invariant, New Moves

We let K be an arc presentation. Then cl(K) is the number of vertical arcs in the diagram. We also have two ways to relate arc presentations of the same knot. Stabilization/Destabilization Moves: There is a simple relation between Reidemeister moves and these new moves!

Untangling the Unknot

New Invariant, New Moves

Exchange Moves There is a simple relation between Reidemeister moves and these new moves!

Untangling the Unknot

New Invariant, New Moves

Exchange Moves Untangling the Unknot

New Invariant, New Moves

Exchange Moves

There is a simple relation between Reidemeister moves and these new moves! Theorem (Dynnikov) Let K be an arc-presentation of the unknot, then there exists a finite sequence of exchange/destabilization moves:

K → K1 → K2 → ... → Kn

such that Kn is trivial.

Untangling the Unknot

Two Key Components of the Theorems on Bounds

Lemma Let K be a knot diagram. Then

cl(K) ≤ 2b(K) + 2cr(K) Untangling the Unknot

Two Key Components of the Theorems on Bounds

Lemma Let K be a knot diagram. Then

cl(K) ≤ 2b(K) + 2cr(K)

Theorem (Dynnikov) Let K be an arc-presentation of the unknot, then there exists a finite sequence of exchange/destabilization moves:

K → K1 → K2 → ... → Kn

such that Kn is trivial. Culprit Reidemeister Moves : 10 Culprit HK : More than 1036 Culprit Lackenby : A little less than 1035

Untangling the Unknot

Looking Ahead

Perhaps we can get better bounds? Culprit Reidemeister Moves : 10 Culprit HK : More than 1036 Culprit Lackenby : A little less than 1035

Untangling the Unknot

Looking Ahead

Perhaps we can get better bounds? Culprit HK : More than 1036 Culprit Lackenby : A little less than 1035

Untangling the Unknot

Looking Ahead

Perhaps we can get better bounds?

Culprit Reidemeister Moves : 10 Culprit Lackenby : A little less than 1035

Untangling the Unknot

Looking Ahead

Perhaps we can get better bounds?

Culprit Reidemeister Moves : 10 Culprit HK : More than 1036 Untangling the Unknot

Looking Ahead

Perhaps we can get better bounds?

Culprit Reidemeister Moves : 10 Culprit HK : More than 1036 Culprit Lackenby : A little less than 1035 Enter width!

Untangling the Unknot

Looking Ahead

Idea: Use another invariant! Untangling the Unknot

Looking Ahead

Idea: Use another invariant! Enter width! Untangling the Unknot

Acknowledgments

Many thanks to Scott for his patience and help! Image sources: http://www.wikipedia.org/ http://arxiv.org/pdf/1006.4176v4.pdf http://arxiv.org/pdf/0903.5543.pdf References: http://arxiv.org/pdf/1006.4176v4.pdf http://arxiv.org/pdf/1302.0180v2.pdf Untangling the Unknot

Thank you for you attention!