yui (結)

Setup

• Install Rust.
• Download yui.
• Run.

$ cargo run -r -- [command] [args]

Optional features

Build with --features [features] to enable optional features. Available options:

• bigint: enables BigInt to avoid overflow.
• poly : enables polynomial types.
• qint : enables quadratic integer types (Gaussian and Eisenstein integers)

e.g.

$ cargo run -r --features bigint,poly -- [command] [args]

Commands

Khovanov homology (kh)

Computes the Khovanov homology for the given link.

Usage: kh <LINK> [OPTIONS]

Arguments:
  <LINK>  

Options:
  -c, --c-value <C_VALUE>  [default: 0]
  -t, --c-type <C_TYPE>    [default: Z]
  -m, --mirror
  -r, --reduced            
  -b, --bigraded (available only when c = 0.)

Examples

$ cargo run -r -- kh 3_1 -b

j\i  -3  -2     -1  0 
 -1                  Z 
 -3                  Z 
 -5       Z           
 -7       (Z/2)       
 -9   Z               

<LINK> can be specified by the pd-code as:

$ cargo run -r -- kh "[[1,4,2,5],[3,6,4,1],[5,2,6,3]]" -b

The following computes the (bigraded) Bar-Natan homology over Q[H].

$ cargo run --features poly -- kh 3_1 -t Q -c H

[-3]: 0
[-2]: (Q[H]/H²)
[-1]: 0
[0]: Q[H]²

Khovanov complex (ckh)

Usage: ckh <LINK> [OPTIONS]

Arguments:
  <LINK>  

Options:
  -c, --c-value <C_VALUE>  [default: 0]
  -t, --c-type <C_TYPE>    [default: Z] [possible values: Z, Q, F2, F3, Gauss, Eisen]
  -m, --mirror             
  -r, --reduced            
  -b, --bigraded           

Examples

$ cargo run -r -- ckh 3_1 -t Z -b

j\i  -3  -2  -1  0 
 -1               Z 
 -3               Z 
 -5       Z        
 -7   Z   Z        
 -9   Z            

C[(-3, -9)]: Z -> 0
[[]]

C[(-3, -7)]: Z -> Z
[[-2]]

...

ckh can be computed over the ring Z[H].

$ cargo run -r --features poly -- ckh 3_1 -t Z -c H

C[-3]: Z[H]² -> Z[H]²
[[-H, -2],
 [0, H]]

C[-2]: Z[H]² -> 0
[[]]

C[-1]: 0 -> Z[H]²
[[]]

C[0]: Z[H]² -> 0
[[]]

ss-invariant (ss)

See this paper for detail.

Usage: ss <LINK> -c <C_VALUE> [OPTIONS]

Arguments:
  <LINK>  

Options:
  -c, --c-value <C_VALUE>  
  -t, --c-type <C_TYPE>    [default: Z] [possible values: Z, Q, F2, F3, Gauss, Eisen]
  -m, --mirror             
  -r, --reduced            

The following computes the s-invariant over Q.

$ cargo run --features poly -- ss 3_1 -t Q -c H
-2

License

yui is released under the MIT license.