Hopf's Umlaufsatz for lattice polygons

Unit lattice step in a given direction to Distinct north-then-west corners sit at distinct vertices

• Unit lattice step in a given direction
• Vertices of a lattice walk from its step directions
• Number of left turns of a direction sequence
• Turns of a cyclic direction sequence sum to zero
• Without right turns, the number of left turns is divisible by four
• A closed walk without right turns turns at least once
• A closed walk without right turns has at least four left turns
• Exactly four left turns when simplicity rules out eight
• Reverse of a dart
• Turn between two directions as an element of \(\mathbb{Z}/4\)
• Integer turn exponent between two directions
• Closed lattice walk starts at the origin
• Forward vertex recursion of a closed lattice walk
• Backward vertex recursion of a closed lattice walk
• Coordinates of the four lattice steps
• North step followed by west step at the top corner of a walk
• North-then-west corner at the highest vertices
• South-then-east corner at the lowest vertices
• Previous and next vertices of a north-then-west corner
• Distinct north-then-west corners sit at distinct vertices

Contiguous top row of a walk to A closed walk crosses each horizontal line an even number of times

• Contiguous top row of a walk
• Recursion for the x-coordinate of a closed walk
• Recursion for the y-coordinate of a closed walk
• No east step leaves a rightmost vertex
• No west step arrives at a rightmost vertex
• No south step leaves a lowest vertex
• No north step arrives at a lowest vertex
• Number of left turns out of a given direction
• Number of straight continuations in a given direction
• Every direction is left by equally many left turns
• Left turns are four times the north-to-west turns
• Eight left turns give two north-then-west corners
• A unique top corner forces fewer than eight left turns
• Vertical lattice direction
• Horizontal lattice direction
• Recursion for the y-coordinate of a walk step
• Recursion for the x-coordinate of a walk step
• Number of crossings of a horizontal line
• A closed walk crosses a horizontal line equally often up and down
• A closed walk crosses each horizontal line an even number of times

A lattice point lies on a walk step to Four left turns, given that two corners force a self-intersection

• A lattice point lies on a walk step
• Crossing steps force a repeated vertex
• Number of crossings of a horizontal ray
• Upward ray crossings above a cell
• Ray-crossing parity of a lattice cell
• Upward crossing count is unchanged across a vertical move
• Ray parity is unchanged across a vertical move
• No upward crossings for cells above a walk
• Cells above a walk are outside
• Upward crossing counts of horizontal neighbours sum to an even number
• Ray parity is unchanged across a horizontal move
• Moving horizontally inside a closed walk
• Moving vertically inside a closed walk
• Coordinates of the vertices around a north-then-west corner
• The north step at a corner blocks horizontal movement
• The west step at a corner blocks vertical movement
• Uniqueness of the lexicographically largest vertex
• Existence of a north-then-west corner
• Two north-then-west corners force a self-intersection
• Four left turns, given that two corners force a self-intersection

Unit cell to the left of a lattice step to \(k + 2 \ne k\) in \(\mathrm{Fin}\, n\) for \(n \ge 3\)

• Unit cell to the left of a lattice step
• Steps around a left turn share their left cell
• No vertical step through a non-vertex point
• No horizontal step starts at a non-vertex point
• Index of a vertical step above a walk vertex
• Index of a horizontal step right of a walk vertex
• Index of a vertical step below a walk vertex
• Index of a horizontal step left of a walk vertex
• Left cells keep their ray parity on a straight step
• Ray parity of the cell left of a step
• Consecutive steps have the same left parity
• Left parity is constant along a walk without right turns
• Vertical steps separating two horizontal neighbour cells
• Parity of upward crossings across a vertical segment
• Ray parity across a vertical segment counts separating steps
• An odd number of vertical steps flips the ray parity
• Horizontal steps on a unit segment
• Upward crossings of vertically adjacent cells differ by separating steps
• An odd number of horizontal steps flips the ray parity
• \(k + 2 \ne k\) in \(\mathrm{Fin}\, n\) for \(n \ge 3\)

Steps on a horizontal unit segment join its endpoints to A closed walk crosses each vertical line an even number of times

• Steps on a horizontal unit segment join its endpoints
• Steps on a vertical unit segment join its endpoints
• A simple closed walk traverses each horizontal segment at most once
• A simple closed walk traverses each vertical segment at most once
• Some north-then-west corner has its below-left cell inside
• No north step leaves a highest vertex
• No south step arrives at a highest vertex
• Position after a run of west steps
• Single west run covering the top row
• A single top run makes the top row contiguous
• Top row is contiguous when every top corner starts a single run
• A north step into a highest vertex is followed by a west step
• A west step on the top row is followed by west or south
• A closed walk is not all west steps
• A west run from a highest vertex ends in a south step
• A west run covering the top row is a single top run
• A step changes the x-coordinate by at most one
• Number of crossings of a vertical line
• A closed walk crosses a vertical line equally often left and right
• A closed walk crosses each vertical line an even number of times

Interior cells by the even-odd rule to Horizontal neighbours of different parity are separated by a vertical step

• Interior cells by the even-odd rule
• Minimal height of a walk
• Maximal height of a walk
• An inside cell lies below a horizontal step
• Cells below a walk are outside
• Finiteness of the interior of a closed walk
• Finite set of interior cells of a closed walk
• Corner count of the interior counts inside cells around a point
• Ray parity is constant around a non-vertex point
• Corner weight at a left turn into the east direction
• Corner weight at a left turn into the north direction
• Corner weight at a left turn into the south direction
• Corner weight at a left turn into the west direction
• A horizontal direction is not vertical
• A vertical direction is not horizontal
• Even corner counts have zero corner weight
• Zero corner weight where the walk continues horizontally
• Zero corner weight where the walk continues vertically
• Straight steps of a simple closed walk carry no corner weight
• Horizontal neighbours of different parity are separated by a vertical step

Vertical neighbours of different parity are separated by a horizontal step to Face-graph adjacency as absence of a walk step between cells

• Vertical neighbours of different parity are separated by a horizontal step
• Interior of a simple closed walk has no pinch points
• Pinch count of the ray-parity interior is zero
• Zero corner weight of the interior off the walk
• Corner defect of the interior as a sum over walk vertices
• Corner weights at the turns of a walk without right turns
• Corner defect of the interior is the number of left turns up to sign
• Four left turns from a pinch-free interior of Euler characteristic 1
• Vertex of a walk as a lattice site
• Vertices of a simple walk are distinct sites
• Each lattice step has \(\ell^1\)-length one
• Consecutive walk vertices are adjacent in the square lattice
• Simple closed walk as an embedded cycle graph
• A cycle graph has one connected component
• Number of edges of a cycle graph
• Cycle graph has cycle rank one
• Subgraph of a simple closed lattice walk has cycle rank one
• Lattice Jordan curve theorem for simple closed walks
• Ray-parity interior equals the bounded face components
• Face-graph adjacency as absence of a walk step between cells

Unit cell as a face of the square lattice to Membership in the bounded face

• Unit cell as a face of the square lattice
• Simple closed walk as a planar subgraph
• Face graph of a closed walk
• Each walk step is an edge of the image graph
• Neighbours in the image graph are walk predecessor and successor
• Horizontally face-adjacent cells have the same ray parity
• Vertically face-adjacent cells have the same ray parity
• Face-adjacent cells have the same ray parity
• Ray parity is constant on face-graph components
• Top corner with an inside cell below it
• Defining property of the inside top corner
• Face below-left of the inside top corner
• The face below the inside top corner is inside
• Inner component of the face graph
• A face far outside the walk has ray parity zero
• The inner face component is finite
• Cells of the inner face component
• The inner component has finitely many cells
• Cells of the bounded face of a simple closed walk
• Membership in the bounded face

Lattice Jordan curve theorem, even-odd form to Zero corner weight of the bounded face off the walk

• Lattice Jordan curve theorem, even-odd form
• Face-graph component of an interior cell
• A side between interior and exterior cells is a walk edge
• Three walk edges cannot meet at a vertex
• Bounded face of a simple closed walk has no pinch points
• Pinch count of the bounded face is zero
• Cell to the left of a walk step
• Cell to the right of a walk step
• Vertices of a walk's image graph are adjacent only to walk neighbours
• Consecutive left cells on a straight step are face-adjacent
• Consecutive left cells have the same ray parity
• All left cells have the same ray parity
• Every left cell has ray parity one
• Every left cell lies in the bounded face
• Cells on either side of a step lie in different faces
• Left and right cells of a step share that step
• No right cell lies in the bounded face
• Corner count as a sum of four indicators
• Corner count of the bounded face at a walk vertex
• Zero corner weight of the bounded face off the walk

Corner weight of the bounded face at a walk vertex to Lower-left corner of a cell as a skeleton vertex

• Corner weight of the bounded face at a walk vertex
• Corner defect of the bounded face equals the number of left turns
• 1-skeleton of a cell complex
• 1-skeleton of a cell complex as a planar subgraph
• Cell complex edges lie in the embedded skeleton
• Shared side of a cell and an adjacent face is an edge
• The face of a cell is enclosed on all four sides
• Face component containing a cell
• The face of a cell is alone in its component
• The face component of a cell is bounded
• Bounded face formed by a cell
• Distinct cells form distinct bounded faces
• Number of cells is at most the nullity of the skeleton
• Edges of the skeleton are the edges of the cell complex
• Vertex count of the skeleton
• Euler characteristic at most 1 for a connected skeleton
• Corner of a cell as a skeleton vertex
• Cell corners are connected to the lower-left corner
• Adjacent cells have connected lower-left corners
• Lower-left corner of a cell as a skeleton vertex

Interior cells have connected lower-left corners to A simple closed walk without right turns has under eight left turns

• Interior cells have connected lower-left corners
• Connectedness of the skeleton of the interior
• Interior of a simple closed walk has Euler characteristic at most 1
• Hopf's Umlaufsatz
• Agreement of the two definitions of the left cell
• A walk without right turns is positively oriented
• Ray-parity interior equals the bounded face
• Discrete Umlaufsatz for walks of length at least three, shifted form
• Discrete Umlaufsatz for walks of length at least three
• A closed walk without right turns has length at least three
• Discrete Umlaufsatz for lattice walks without right turns
• A simple closed walk without right turns has under eight left turns

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