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➰ A program for manipulating and playing with knot diagrams.

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# knots
➰ A program for manipulating and playing with knot diagrams.

## Description
### Grid Diagrams
One interesting way of representing a knot is via a so-called `n`x`n` _grid diagram_. Each cell of the grid is either an `x`, an `o`, or "blank." Each row and column is restricted to have _exactly_ one `x` and one `o`. We can construct the knot corresponding to a particular grid diagram using the following procedure:

1. In each column, connect each `x` to the corresponding `o` 
2. In each row, connect each `o` to the corresponding `x`
3. Whenever a horizontal segment intersects a vertical segment, assume that the vertical segment passes _over_ the horizontal segment (i.e. a grid diagram _only_ consists of over-crossings)

### Refinement
Following the procedure outlined above results in a piecewise linear link (polyline). To obtain a "smoother" projection of the knot (one without sharp corners), we need to perform some form of topological refinement, taking care to not change the underlying structure of the knot. Following Dr. Scharein's thesis (link below), each vertex of the generated polyline is treated as a particle in a physics simulation. Adjacent vertices are attracted to one another via a mechanical spring force. Non-adjacent particles are repelled from one another via an electrostatic force, which (in practice) prevents segments from crossing over or under one another. A more robust method would perform intersection tests between all pairs of non-neighboring line segments to prevent any "illegal" crossings. This is currently a TODO item.

In order for this to work, the polyline must start in a "valid" state, where no two line segments intersect. To accomodate this, crossings are calculated based on criteria `(3)` above. A new vertex is added at each point of crossing and "lifted" along the `z`-axis by some small amount. The entire polyline is then resampled, to increase vertex density.

Before the knot is rendered, a path-guided extrusion is performed to "thicken" the knot. At each vertex along the polyline, a coordinate frame is established by calculating the tangent vector and a vector orthogonal to the tangent. Then, a circular cross-section is added at the origin of this new, local coordinate system. Adjacent cross-sections are connected with triangles to form a continuous, closed "tube." To avoid jarring rotations, [parallel transport](https://en.wikipedia.org/wiki/Parallel_transport) is employed. Essentially, each successive coordinate frame is calculated with respect to the previous frame. This ensures that the circular cross-sections smoothly rotate around the polyline during traversal.

### Cromwell Moves
The Cromwell Moves are similar to the [Reidemeister Moves](https://en.wikipedia.org/wiki/Reidemeister_move), specifically applied to grid diagrams. They all us to obtain isotopic knots, i.e. knots that have the same underlying topology but "look" different. This gives us a way to systematically explore a given knot invariant.

In `knots`, the Cromwell Moves are represented as an `enum`:

```rust
enum CromwellMove {
    // A move that cyclically translates a row or column in one of four directions: up, down, left, or right
    Translation(Direction),

// A move that exchanges to adjacent, non-interleaved rows or columns
    Commutation { axis: Axis, start_index: usize, },

// A move that replaces an `x` with a 2x2 sub-grid
    Stabilization { cardinality: Cardinality, i: usize, j: usize, },
    
    // A move that replaces a 2x2 sub-grid with an `x` (the opposite of an x-stabilization): currently not supported
    Destabilization { cardinality: Cardinality, i: usize, j: usize, },
}
```

## Tested On
- Windows 8.1
- NVIDIA GeForce GT 750M
- Rust compiler version `1.38.0-nightly` (nightly may not be required)

NOTE: this project will only run on graphics cards that support OpenGL [Direct State Access](https://www.khronos.org/opengl/wiki/Direct_State_Access) (DSA).

## To Build
1. Clone this repo.
2. Make sure 🦀 [Rust](https://www.rust-lang.org/en-US/) installed and `cargo` is in your `PATH`.
3. Inside the repo, run: `cargo build --release`.

## To Use
All grid diagrams must be "square" `.csv` files (the same number of rows as columns). Each row and column must have _exactly_ one `x` and one `o`: all other entries should be spaces ("blank"). The grid diagram will be validated upon construction, but the program will `panic!` if one of the conditions above is not met. An example grid diagram for the trefoil knot is shown below:

```
"x"," ","o"," "," "
" ","x"," ","o"," "
" "," ","x"," ","o"
"o"," "," ","x"," "
" ","o"," "," ","x"
```

To rotate the camera around the object in 3-dimensions, press + drag the left mouse button. Press `h` to "home" (i.e. reset) the camera.

You can change between wireframe and filled modes by pressing `w` and `f`. You can save out a screenshot by pressing `s`. Finally, you can reset the physics simulation by pressing `r`.

## To Do
- [ ] Implement a knot "drawing" tool
- [ ] Add segment-segment intersection test for more robust topological refinement
- [ ] Add a GUI for viewing the current grid diagram
- [ ] Generate the Dowker notation for a given knot diagram
- [ ] Explore planar graphs and their relationship to knot diagrams
- [ ] Implement a `tangle` calculator for generating knots (Conway notation)

## Credits
This project was largely inspired by and based on previous work done by Dr. Robert Scharein, whose PhD [thesis](https://knotplot.com/thesis/) and [software](https://knotplot.com/) were vital towards my understanding of the relaxation / meshing procedures.

### License
[Creative Commons Attribution 4.0 International License](https://creativecommons.org/licenses/by/4.0/)

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