What it does
A braid is a set of strands running down the page that cross each other. Any braid can be written as a word in the generators σ1, σ2, …: σi crosses the strands in places i and i + 1, and σi⁻¹ crosses them the other way. Joining the bottom of each strand back to its top gives the closure, which is a knot or a link. By Alexander’s theorem, every knot can be drawn this way.
This tool reads a braid word and:
- draws the braid, crossing by crossing, with the over-strand shown unbroken, and its closure, with each component in its own colour;
- counts the crossings and the writhe (exponent sum: +1 for each σi, −1 for each σi⁻¹);
- gives the permutation of the strands and the number of components of the closure (one for each cycle of the permutation);
- for links, the linking number of each pair of components;
- works out the Alexander polynomial Δ(t) of the closure exactly, plus the knot determinant |Δ(−1)| and a lower bound for the genus;
- says which knot in the named table has the same Δ.
How to use it
- Choose a named knot, or type your own braid word. Write it as
σ1 σ2^-1,s1 s2^-1,1 -2or with lettersa B(a = σ1, A = σ1⁻¹, b = σ2, …). Powers and brackets work:(σ1 σ2)^4. - The number of strands is one more than the largest index. Type a larger number to add strands that do not cross (each adds a separate unknot to the closure).
- Switch the closure and component colours on or off. Download the drawing and facts as an image, PDF or Word file.
How the Alexander polynomial is found
Each σi is sent to an (n − 1) × (n − 1) matrix with entries in t (the reduced Burau representation), and the braid to the product of these matrices B. Then Δ(t) is det(I − B) × (1 − t)/(1 − t^n), computed with exact whole-number polynomials. It is only defined up to multiplying by ±t^k, so for a knot the page writes it symmetric with Δ(1) = 1, and for a link with lowest power t^0.
Worked example
The page opens with the figure-eight knot, σ1 σ2⁻¹ σ1 σ2⁻¹ on 3 strands. It has 4 crossings and writhe 0, the permutation is the 3-cycle (1 2 3), so the closure is a knot. Its Alexander polynomial is −t + 3 − t^-1, the determinant is 5 and the genus is at least 1. The trefoil σ1^3 gives t − 1 + t^-1 (determinant 3), and the Hopf link σ1^2 has 2 components with linking number 1 and Δ = t − 1. The Borromean rings (σ1 σ2⁻¹)^3 have 3 components, every pair has linking number 0, yet Δ = t^4 − 4t^3 + 6t^2 − 4t + 1, so they are linked.
Frequently asked questions
Does the same Alexander polynomial mean the same knot? No. Different knots can share Δ (for example, some knots have Δ = 1 like the unknot). A match is a hint, not a proof.
Why is the crossing count not the knot’s crossing number? A braid usually uses more crossings than the simplest diagram of its closure. The figure-eight needs 4 either way, but many knots need extra crossings as a braid.
Is anything sent anywhere? No. Everything runs in your browser.
Where the numbers come from
- Standard knot theory: Artin's braid group, Alexander's theorem (every knot is a closed braid) and the Burau formula Δ(t) ≐ det(I − B(β)) (1 − t)/(1 − t^n). The braid words of the named knots are standard ones and are mathematical facts.
- Checked against sympy, which worked the Alexander polynomials out a different way (unreduced Burau matrix, deleting a row and column), for the named knots and 25 random braids; the named values match the usual knot tables.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.