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Enumeration of knot diagrams usingconstraint programming

Ciaran McCreesh, Alice Miller, Patrick Prosser,Craig Reilly, James Trimble

What is a knot?

A knot is an embedding of the circle in R3.

An intuitive way to think about this is to consider a knot as aknotted piece of string with the ends glued together.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 1 / 18

What is a knot?

A knot is an embedding of the circle in R3.An intuitive way to think about this is to consider a knot as aknotted piece of string with the ends glued together.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 1 / 18

Drawing knots on paper

A function f : R3 R2 where f (x , y , z) = f (x , y), is called aprojection map, and the image of a knot K under f is calledthe projection of K.

Such a projection is often referred to as the shadow of K.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 2 / 18

Drawing knots on paper

Information regarding the orientation of arcs at crossings isgiven by leaving gaps in a knots shadow.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 2 / 18

Drawing knots on paper

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 2 / 18

Drawing knots on paper

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 2 / 18

Prime knots

Prime knots are knots which cannot be decomposed. Fornontrivial knots this means that they cannot be written as theconnect sum of two non-trivial knots.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 3 / 18

Prime knots

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 3 / 18

Equivelent knots and Reidemeister Moves

When enumerating prime knots it is convention to give eachprime knot by a knot diagram which has as few as possiblecrossings.

Two knots K and J are equivalent if K can be transformed intoJ by a series of elementary deformations.

Showing that two knots are equivalent by elementarydeformations is not practical.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 4 / 18

Equivelent knots and Reidemeister Moves

In the 1926 Reidermister proved that two knot diagrams k0 andk1 of the same knot K can be related by a sequence of thefollowing three moves (called Reidermeister moves):1

1Alexander and Briggs developed these moves independently in 1927Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 4 / 18

Equivelent knots and Reidemeister Moves

In the 1926 Reidermister proved that two knot diagrams k0 andk1 of the same knot K can be related by a sequence of thefollowing three moves (called Reidermeister moves):1

1Alexander and Briggs developed these moves independently in 1927Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 4 / 18

Representations of knots

Knot diagrams are really just 4-valent planar graphs.

The nodes in the graph correspond to the crossings in the knotdiagram.The edges between nodes correspond to arcs between crossingsin the knot diagram.The arcs are labelled with their orientation at their source andtarget crossings.

Other structures familiar to computing scientists can be used,linked lists of crossings were popular in the 1950s.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 5 / 18

Representations of knots

The representations used by topologists are typically also usedfor representing knots in a computer.

Examples are Dowker-Thistlethwait codes (DT codes), Gausscodes, braid representatives, Conway notation, and many more.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 5 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

knot, writing out a list of thenumbers you come to (with anegative sign indicating that acrossing was visited on an understrand). Stop when each numberappears twice once with each sign.

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

knot, writing out a list of thenumbers you come to (with anegative sign indicating that acrossing was visited on an understrand). Stop when each numberappears twice once with each sign.

1

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

knot, writing out a list of thenumbers you come to (with anegative sign indicating that acrossing was visited on an understrand). Stop when each numberappears twice once with each sign.

1 2

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

1 2

3

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

1 2

3

4

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

1 2

3

4

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around the

1 2

3

4

left

Ciaran McCreesh, Alice Miller, Brendan Owens, Patrick Prosser, Craig Reilly, James Trimble

Enumeration of knot diagrams using constraint programming 6 / 18

Representing knots with Gauss codes

The strategy for representing a givenknot (with n crossings) by a Gausscode is as follows.

1 Label the crossings with the numbers1 to n.

2 Pick a point on the knot.3 Pick a direction and walk around