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Unknotted Cycles Nathan McNew Towson University Joint with - - PowerPoint PPT Presentation

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Unknotted Cycles Nathan McNew Towson University Joint with Christopher Cornwell Permutation Patterns 2018 Dartmouth College Hanover, NH July 13th, 2018 Nathan McNew Unknotted Cycles July 13th, 2018 1 Cycle Diagrams The cycle diagram of a

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Unknotted Cycles

Nathan McNew Towson University Joint with Christopher Cornwell Permutation Patterns 2018 Dartmouth College Hanover, NH July 13th, 2018

Nathan McNew Unknotted Cycles July 13th, 2018 1

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)).

Nathan McNew Unknotted Cycles July 13th, 2018 2

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example:

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example: σ = 467513298

Nathan McNew Unknotted Cycles July 13th, 2018 2

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example: σ = 467513298

Nathan McNew Unknotted Cycles July 13th, 2018 2

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example: σ = 467513298

Nathan McNew Unknotted Cycles July 13th, 2018 2

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example: σ = 467513298 = (145)(2637)(89)

Nathan McNew Unknotted Cycles July 13th, 2018 2

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Cycle Diagrams

The cycle diagram of a permutation σ of length n is drawn on an n×n grid by plotting each point (i, σ(i)) and a vertical line from (i, i) to (i, σ(i)), followed by a horizontal line from (i, σ(i)) to (σ(i), σ(i)). Example: σ = 467513298 Note: The point (i, σ(i)) is a corner as is every (i, i) on the diagonal.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other. The unknot is any knot that can be so deformed to a circle.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other. The unknot is any knot that can be so deformed to a circle.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other. The unknot is any knot that can be so deformed to a circle. A link is an embedding of one or more circles into R3.

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other. The unknot is any knot that can be so deformed to a circle. A link is an embedding of one or more circles into R3. (A collection of knots, which may be “linked” to each other.)

Nathan McNew Unknotted Cycles July 13th, 2018 3

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Grid Diagrams

A Grid Diagram is a way to present a knot (or link) on an n×n grid. A knot is an embedding of a circle into 3 dimensional space. Two knots are considered “the same” if one can be continuously deformed (without self-intersecting) to produce the other. The unknot is any knot that can be so deformed to a circle. A link is an embedding of one or more circles into R3. (A collection of knots, which may be “linked” to each other.) The unlink is the link in which every component is the unknot, and none of the components are themselves ”linked.”

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Grid Diagrams

A grid diagram is an n×n grid with exactly two points in each row

  • r column and a line connecting the entries in each row or column.

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Grid Diagrams

A grid diagram is an n×n grid with exactly two points in each row

  • r column and a line connecting the entries in each row or column.

The diagram is interpreted as a knot (or link) by designating all of the vertical lines to cross over the horizontal lines.

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Grid Diagrams

A grid diagram is an n×n grid with exactly two points in each row

  • r column and a line connecting the entries in each row or column.

The diagram is interpreted as a knot (or link) by designating all of the vertical lines to cross over the horizontal lines.

Nathan McNew Unknotted Cycles July 13th, 2018 4

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Grid Diagrams

A grid diagram is an n×n grid with exactly two points in each row

  • r column and a line connecting the entries in each row or column.

The diagram is interpreted as a knot (or link) by designating all of the vertical lines to cross over the horizontal lines.

Nathan McNew Unknotted Cycles July 13th, 2018 4

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Grid Diagrams

A grid diagram is an n×n grid with exactly two points in each row

  • r column and a line connecting the entries in each row or column.

The diagram is interpreted as a knot (or link) by designating all of the vertical lines to cross over the horizontal lines.

Nathan McNew Unknotted Cycles July 13th, 2018 4

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Grid Diagrams

Fun facts about grid diagrams:

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Grid Diagrams

Fun facts about grid diagrams: Every knot can be represented as a grid diagram.

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Grid Diagrams

Fun facts about grid diagrams: Every knot can be represented as a grid diagram. There are rules called Cromwell moves that act on grid diagrams without changing the knot type that can transform a grid diagram of a knot into any other grid diagram of the same knot.

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Observation

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Observation

Grid diagrams are remarkably similar to cycle diagrams.

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Observation

Grid diagrams are remarkably similar to cycle diagrams. Treating diagonal entries of a cycle diagram as points the only differences are:

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Observation

Grid diagrams are remarkably similar to cycle diagrams. Treating diagonal entries of a cycle diagram as points the only differences are: In cycle diagrams one of the two points in each row/column must lie on the diagonal, which isn’t the case for grid diagrams.

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Observation

Grid diagrams are remarkably similar to cycle diagrams. Treating diagonal entries of a cycle diagram as points the only differences are: In cycle diagrams one of the two points in each row/column must lie on the diagonal, which isn’t the case for grid diagrams. In grid diagrams, it is not allowed to have a single point in a row

  • r column as occur in cycle diagrams when there are fixed points.

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Observation

Grid diagrams are remarkably similar to cycle diagrams. Treating diagonal entries of a cycle diagram as points the only differences are: In cycle diagrams one of the two points in each row/column must lie on the diagonal, which isn’t the case for grid diagrams. In grid diagrams, it is not allowed to have a single point in a row

  • r column as occur in cycle diagrams when there are fixed points.

Restricting our attention to derangements, permutations without fixed points, every cycle diagram can be interpreted as a grid diagram, associating a link (knot) to each derangement (cycle).

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The link associated to a permutation

Definition

For any derangement σ, the link associated to σ is obtained by drawing the cycle diagram of σ and interpreting it as a grid diagram

  • instead. If σ is a cycle then this is the knot associated to σ.

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The link associated to a permutation

Definition

For any derangement σ, the link associated to σ is obtained by drawing the cycle diagram of σ and interpreting it as a grid diagram

  • instead. If σ is a cycle then this is the knot associated to σ.

We will call any cycle associated to the unknot an unknotted cycle.

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

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Example: σ = 837295641

... the unknot!

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Example: σ = 57428163

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Example: σ = 57428163

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Example: σ = 57428163

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Example: σ = 57428163

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Example: σ = 57428163

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Example: σ = 57428163

... a trefoil knot.

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Counting Unknotted Cycles

How many cycles of length n are unknotted?

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles 2 1

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles 2 1 3 2

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6 5 22 · · ·

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Counting Unknotted Cycles

How many cycles of length n are unknotted? n # cycles 2 1 3 2 4 6 5 22 · · · 6 90 · · ·

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Counting Unknotted Cycles

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Counting Unknotted Cycles

Denote by Sn the nth large Schr¨

  • der number, given by the recurrence

S1 = 1 and Sn = Sn−1 +

n−1

  • k=1

SkSn−k.

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Counting Unknotted Cycles

Denote by Sn the nth large Schr¨

  • der number, given by the recurrence

S1 = 1 and Sn = Sn−1 +

n−1

  • k=1

SkSn−k.

Theorem

The count of unknotted cycles of size n+1 is Sn.

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Rooted-Signed-Binary-Trees

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Rooted-Signed-Binary-Trees

Definition

A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative.

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Rooted-Signed-Binary-Trees

Definition

A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations.

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Rooted-Signed-Binary-Trees

Definition

A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either:

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Rooted-Signed-Binary-Trees

Definition

A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either: A child node can be rotated into a parent with the same sign.

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Rooted-Signed-Binary-Trees

Definition

A rooted-signed-binary tree is a rooted binary tree where each non-root node is either positive or negative. Two trees are equivalent if one can be obtained from another by a series of tree rotations. The allowed tree rotations are either: A child node can be rotated into a parent with the same sign. A node can be rotated into the root. The new node is given the sign of the node rotated into the root.

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Example: Rooted-signed-binary-trees

+ + − −

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Example: Rooted-signed-binary-trees

+ + − − + + − −

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Example: Rooted-signed-binary-trees

+ + − − + + − − + + − −

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The bijection

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type.

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted.

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21.

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the ith position of the tree:

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the ith position of the tree: For a positive node insert i+1 prior to the element in position i. Increment each element in the cycle that had value i+1 or more.

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The bijection

Key observation: Inserting a point in a cycle diagram immediately above or below the diagonal does not affect the knot type. Use the signed tree to keep track of how points are inserted. Assign the tree with only the unsigned root to the cycle σ = 21. Number the places a leaf could be added to the tree. To add a node to the tree into the ith position of the tree: For a positive node insert i+1 prior to the element in position i. Increment each element in the cycle that had value i+1 or more. For a negative node insert i after the element in position i. Increment each element in the cycle that had value i or greater.

Nathan McNew Unknotted Cycles July 13th, 2018 14

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The bijection

Before insertion After inserting ⊕ After inserting ⊖

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Example

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Example

+ + − −

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Example

+ + − −

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Example

+ + − −

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Example

+ + − −

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Example

+ + − − +

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Example

+ + − − +

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Example

+ + − − + + +

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Example

+ + − − + + +

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Example

+ + − − + + + + + −

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Example

+ + − − + + + + + −

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Example

+ + + + + − + + − −

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Example

+ + + + + − + + − −

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Ideas of proof

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation: Map from trees to cycles is injective:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation: Map from trees to cycles is injective:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation: Map from trees to cycles is injective: Map from trees to cycles is surjective:

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation: Map from trees to cycles is injective: Map from trees to cycles is surjective: ...not so easy.

Nathan McNew Unknotted Cycles July 13th, 2018 17

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Ideas of proof

Show large Scr¨

  • der numbers count rooted-signed-binary-trees:

Well-defined as to the order vertices are added to a tree: Well-defined on equivalent trees after a rotation: Map from trees to cycles is injective: Map from trees to cycles is surjective: ...not so easy. To show surjectivity, it would suffice to show that every unknotted cycle, σ, has a point on the off-diagonal, i.e |σ(i) − i| = 1.

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Topology

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Topology

Let σ be a cycle.

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Topology

Let σ be a cycle. Let C(σ) count crossings in the cycle diagram of σ.

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Topology

Let σ be a cycle. Let C(σ) count crossings in the cycle diagram of σ. Note: we get a crossing in σ for each pair of indices (i,j) with either i < j < σ(i) < σ(j)

  • r

i > j > σ(i) > σ(j).

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Topology

Let σ be a cycle. Let C(σ) count crossings in the cycle diagram of σ. Note: we get a crossing in σ for each pair of indices (i,j) with either i < j < σ(i) < σ(j)

  • r

i > j > σ(i) > σ(j). In either case, the crossing is a negative crossing. Let UR(σ) count the number of upper right corners in the diagram:

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Topology

Let σ be a cycle. Let C(σ) count crossings in the cycle diagram of σ. Note: we get a crossing in σ for each pair of indices (i,j) with either i < j < σ(i) < σ(j)

  • r

i > j > σ(i) > σ(j). In either case, the crossing is a negative crossing. Let UR(σ) count the number of upper right corners in the diagram: σ−1(i) < i and σ(i) < i.

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Topology

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Topology

Theorem (Bennequinn’s Innequality)

Let σ be a cycle of length at least 2, K the knot associated to σ, and g(K) the Seifert genus of K. Then C(σ) − UR(σ) ≤ 2g(K) − 1.

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Topology

Theorem (Bennequinn’s Innequality)

Let σ be a cycle of length at least 2, K the knot associated to σ, and g(K) the Seifert genus of K. Then C(σ) − UR(σ) ≤ 2g(K) − 1.

Lemma

If σ is a cycle with |σ(i) − i| ≥ 2 for all i, then each upper right corner of σ corresponds to a unique crossing of σ. So C(σ) ≥ UR(σ).

Theorem

If σ is an unknotted cycle, then |σ(i) − i| = 1 for at least one index i. Proof:

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Topology

Theorem (Bennequinn’s Innequality)

Let σ be a cycle of length at least 2, K the knot associated to σ, and g(K) the Seifert genus of K. Then C(σ) − UR(σ) ≤ 2g(K) − 1.

Lemma

If σ is a cycle with |σ(i) − i| ≥ 2 for all i, then each upper right corner of σ corresponds to a unique crossing of σ. So C(σ) ≥ UR(σ).

Theorem

If σ is an unknotted cycle, then |σ(i) − i| = 1 for at least one index i. Proof: If K is an unknot, then g(K) = 0.

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Topology

Theorem (Bennequinn’s Innequality)

Let σ be a cycle of length at least 2, K the knot associated to σ, and g(K) the Seifert genus of K. Then C(σ) − UR(σ) ≤ 2g(K) − 1.

Lemma

If σ is a cycle with |σ(i) − i| ≥ 2 for all i, then each upper right corner of σ corresponds to a unique crossing of σ. So C(σ) ≥ UR(σ).

Theorem

If σ is an unknotted cycle, then |σ(i) − i| = 1 for at least one index i. Proof: If K is an unknot, then g(K) = 0. So C(σ) − UR(σ) ≤ −1.

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Topology

Corollary

The map from rooted-signed-binary trees to unknotted cycles is surjective.

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Unlinks

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Unlinks

In fact, in our situation, Bennequinn’s innequality is an equality. C(σ) − UR(σ) = 2g(K) − 1.

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Unlinks

In fact, in our situation, Bennequinn’s innequality is an equality. C(σ) − UR(σ) = 2g(K) − 1.

Corollary

If σ is an unlinked derangement then no crossing in the cycle diagram

  • f σ is between different components of the link.

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Unlinks

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Unlinks

The only way to get an unlink is to have unknotted components next to each other, or embedded inside of each other without crossing.

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Unlinks

The only way to get an unlink is to have unknotted components next to each other, or embedded inside of each other without crossing. σ = 845327691

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Counting Unlinks

Theorem

Let U be the set of unlinked permutations (derangements), and denote by c(σ) the number of cycles (knots) in σ.

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Counting Unlinks

Theorem

Let U be the set of unlinked permutations (derangements), and denote by c(σ) the number of cycles (knots) in σ. Define F(u, x) = 1 +

  • σ∈U

uc(σ)x|σ|.

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Counting Unlinks

Theorem

Let U be the set of unlinked permutations (derangements), and denote by c(σ) the number of cycles (knots) in σ. Define F(u, x) = 1 +

  • σ∈U

uc(σ)x|σ|. Then F(u, x) satisfies the recurrence

(2−ux)F(u, x)+ux2F(u, x)2+uxF(u, x)

  • 1−6xF(u, x)+x2F(u, x)2

2 = 1

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Counting Unlinks

Theorem

Let U be the set of unlinked permutations (derangements), and denote by c(σ) the number of cycles (knots) in σ. Define F(u, x) = 1 +

  • σ∈U

uc(σ)x|σ|. Then F(u, x) satisfies the recurrence

(2−ux)F(u, x)+ux2F(u, x)2+uxF(u, x)

  • 1−6xF(u, x)+x2F(u, x)2

2 = 1

  • r equivalently

1 + (ux − 2)F(u, x) + (1 − ux − ux2)F(u, x)2 + (ux2 + u2x3)F(u, x)3 = 0.

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Counting Unlinks

Observations: The coefficient of u in F(u, x) is 1

2(x−x2−x

√ x2−6x+1), the (shifted) generating function of the large Schr¨

  • der numbers.

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Counting Unlinks

Observations: The coefficient of u in F(u, x) is 1

2(x−x2−x

√ x2−6x+1), the (shifted) generating function of the large Schr¨

  • der numbers.

Set u=1 to get the generating function F(x) for the sequence 1, 2, 8, 32, 143, 674, 3316, 16832... of all unlinked permutations.

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Counting Unlinks

Observations: The coefficient of u in F(u, x) is 1

2(x−x2−x

√ x2−6x+1), the (shifted) generating function of the large Schr¨

  • der numbers.

Set u=1 to get the generating function F(x) for the sequence 1, 2, 8, 32, 143, 674, 3316, 16832... of all unlinked permutations. (2 − x)F(x) + x2F(x)2 + xF(x)

  • 1 − 6xF(x) + x2F(x)2

2 = 1

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Open Questions

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Open Questions

Count any knot/link type besides the unknot/unlink.

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . .

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal):

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal): 0,0,0,2,10,34,96,248 . . .

What is the most common knot type for a given cycle length?

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal): 0,0,0,2,10,34,96,248 . . .

What is the most common knot type for a given cycle length? Unlink dominates for 2≤n≤8, trefoil dominates for n=9, 10, 11...

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal): 0,0,0,2,10,34,96,248 . . .

What is the most common knot type for a given cycle length? Unlink dominates for 2≤n≤8, trefoil dominates for n=9, 10, 11... Can we classify all knots that are associated to some cycle?

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal): 0,0,0,2,10,34,96,248 . . .

What is the most common knot type for a given cycle length? Unlink dominates for 2≤n≤8, trefoil dominates for n=9, 10, 11... Can we classify all knots that are associated to some cycle?

Must be negative knots.

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Open Questions

Count any knot/link type besides the unknot/unlink.

Trefoils: 0,0,0,2,28,264,2098, 15176 . . . It would suffice to count primitive trefoil cycles (those without points on the off diagonal): 0,0,0,2,10,34,96,248 . . .

What is the most common knot type for a given cycle length? Unlink dominates for 2≤n≤8, trefoil dominates for n=9, 10, 11... Can we classify all knots that are associated to some cycle?

Must be negative knots. So far only negative braid knots and connected sums of negative braid knots have been observed.

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Thank you!

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