Convert braid words into unitary matrices and study trace invariants.
- Implemented σ₁ and σ₂ generators
- Computed braid word matrices
- Verified trace invariance for trefoil knot
- Introduce θ = γ log(p)
- Explore parameterized invariants
- Move toward Kauffman bracket / Jones polynomial
Bilal Khan
- Generator Matrices Output
When running generators.py, we get the braid matrices σ₁ and σ₂:
[[ 0.5 +0.j -0.8660254+0.j 0. +0.j 0. +0.j] [ 0.8660254+0.j 0.5 +0.j 0. +0.j 0. +0.j] [ 0. +0.j 0. +0.j 1. +0.j 0. +0.j] [ 0. +0.j 0. +0.j 0. +0.j 1. +0.j]] [[ 1. +0.j 0. +0.j 0. +0.j 0. +0.j] [ 0. +0.j 0.5 +0.j -0.8660254+0.j 0. +0.j] [ 0. +0.j 0.8660254+0.j 0.5 +0.j 0. +0.j] [ 0. +0.j 0. +0.j 0. +0.j 1. +0.j]] Simple meaning:
These matrices represent braid crossings as rotations.
- Braid Word Output
For trefoil knot [1, 2, 1]:
Final matrix for trefoil (first 4x4): [[ 0.5 +0.j -0.8660254+0.j 0. +0.j 0. +0.j] [ 0. +0.j 0. +0.j 1. +0.j 0. +0.j] [ 0. +0.j 0. +0.j 0. +0.j 1. +0.j]] Meaning:
A braid word is converted into a final matrix using multiplication.
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Trace Results First run: Trace of trefoil braid: 7 Magnitude: 7.000000 Later run (after parameter update): Trace of trefoil braid: 7.7190269566 Magnitude: 7.719027
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Invariance Test Same knot, different word: [1,2,1] and [2,1,2]
Magnitudes match: True Meaning:
Different braid representation gives same invariant value.
- Experiment Results [1, 2, 1] → 7.719 [2, 1, 2] → 7.719 [1, 2, 1, 2] → 7.719 [1, -1] → 7.719 [1, 1, 2] → 7.719 Observation:
All tested braid words produced the same trace magnitude in current model.
The computed trace magnitude remains constant across different braid words, including both Trefoil (T(2,3)) and Cinquefoil (T(2,5)).
This indicates that the current matrix representation produces a stable but non-discriminative invariant.
While the system successfully processes braid words of varying length, it does not yet distinguish between different knot structures.
This suggests that further refinement is required to obtain a more informative topological invariant.
race of trefoil braid: (7.719026956613581+0j) Magnitude of trace: 7.719027
Same knot, different word: trace = (7.719026956613581+0j), magnitude = 7.719027 Magnitudes match: True
Cinquefoil results: Trace: (7.719026956613581+0j) Magnitude: 7.719027
--- Scaling Test --- Trefoil → 7.719027 Cinquefoil → 7.719027