Mathematics of knots
To discuss topology in physics, you first need the mathematical tools. Here is the minimal toolkit: isotopy, Reidemeister moves, invariants, the Jones polynomial.
1. What is a knot, mathematically?
A knot is formally a closed curve embedded in three-dimensional space that never crosses itself. In mathematical notation:
$$K : S^1 \hookrightarrow \mathbb{R}^3$$where $S^1$ is the circle (the starting "piece of string") and $\mathbb{R}^3$ is ordinary space. The symbol $\hookrightarrow$ means "embedding": a continuous injection with no self-intersection.
Two knots are considered equivalent if one can be deformed continuously into the other without cutting or passing the string through itself. This equivalence is called ambient isotopy.
2. Representation: knot diagrams
In practice, we work with projections onto the plane: we draw the knot seen from above, indicating at each crossing which strand passes over the other (with a small break in the under-strand).
3. The Reidemeister moves
Fundamental question: how can we tell whether two diagrams represent the same knot (up to isotopy)? Kurt Reidemeister's answer (1927): it is enough to check that you can pass from one to the other through a succession of three elementary moves:
- R1: add or remove a loop
- R2: add or remove a pair of crossings
- R3: slide a strand over a crossing
Reidemeister's theorem: two diagrams represent the same knot if and only if one can be obtained from the other by a finite sequence of R1, R2, R3 moves.
Important consequence: to show that a quantity $V(K)$ computed from the diagram is an invariant of the knot, it suffices to check that it is unchanged under R1, R2, R3.
4. A few useful invariants
The crossing number $c(K)$
This is the minimal number of crossings over all possible diagrams of the knot. For the trefoil, $c(3_1) = 3$. For the figure-eight knot, $c(4_1) = 4$. It is a non-negative integer.
The braid index $b(K)$
The minimal number of strands needed to close the knot as a braid. For the trefoil, $b(3_1) = 2$. It is also the minimal number of generators $\sigma_i$ needed to build it.
Chirality
Some knots (like the trefoil) are different from their mirror image. They are called chiral. Others (like the figure-eight 4₁) are identical to their mirror image: they are called amphichiral. This distinction is essential in TQNT: antiparticles are the mirror images of particles.
5. The Jones polynomial $V_K(t)$
The central invariant of TQNT, discovered by Vaughan Jones in 1984, which earned him the Fields Medal in 1990. It is a Laurent polynomial in $t^{1/2}$ that can be computed from the diagram by a simple rule, the skein relation:
$$t^{-1}\, V_{L_+}(t) - t\, V_{L_-}(t) = (t^{1/2} - t^{-1/2})\, V_{L_0}(t)$$where $L_+$, $L_-$, $L_0$ are three knots that differ at exactly one crossing (positive, negative, or smoothed). For the unknot, we set $V_{0_1}(t) = 1$. The skein rule then lets us compute $V_K(t)$ for any knot, step by step.
A few examples:
| Knot | $V_K(t)$ |
|---|---|
| $0_1$ | $1$ |
| $3_1$ (right-handed trefoil) | $-t^{-4} + t^{-3} + t^{-1}$ |
| $3_1^*$ (left-handed trefoil) | $-t^4 + t^3 + t$ |
| $4_1$ | $t^{-2} - t^{-1} + 1 - t + t^2$ |
| $5_1$ | $-t^{-7} + t^{-6} - t^{-5} + t^{-4} + t^{-2}$ |
Observation 1: $V_K$ distinguishes the right-handed trefoil from its mirror image. For amphichiral knots like the figure-eight, $V_K(t) = V_K(t^{-1})$: the polynomial is invariant under $t \leftrightarrow t^{-1}$.
Observation 2: the Jones polynomial captures a lot of information about the knot, but not all of it. There exist pairs of distinct knots with the same Jones polynomial (the Kanenobu pairs). In TQNT, we complement it with other invariants.
6. Complexity index $c_J$ (used in TQNT)
TQNT defines a topological complexity index from the Jones polynomial:
$$c_J(K) = \text{span}(V_K) + \gamma\, |V_K(-1)|$$where $\text{span}(V_K)$ is the difference between the largest and smallest exponent of $V_K(t)$, and $V_K(-1)$ is the polynomial evaluated at $t = -1$ (also called the determinant of the knot). The parameter $\gamma = 0.1$ belongs to the microscopic layer of the χ solver — never fitted to observables, outside the count of the 10 global parameters.
A few values:
- $c_J(3_1) \approx 3.3$ (trefoil — the electron)
- $c_J(5_1) \approx 5.5$
- $c_J(7_2) \approx 8.1$
- $c_J(8_3) \approx 10.2$ (used in the W, Z, H, top bosons)
The larger $c_J$ is, the more topologically complex the knot — and the heavier the corresponding particle (see Module 4).
7. Why TQNT needs all of this
Topological invariants (crossing number, braid index, Jones polynomial, chirality…) are rigid features of a knot: they do not change when the knot is deformed without being cut. They are perfect for representing physical properties that are quantized and stable: charge, spin, mass, stability.
That is why TQNT identifies each particle with a knot and reads off its physical properties from its topological invariants. A particle is not an arbitrary point — it is a precise geometric shape, and that shape encodes everything you need to know.
A knot is a closed curve in $\mathbb{R}^3$, studied up to isotopy. Reidemeister's theorem gives the elementary rules to identify two knots. The Jones polynomial distinguishes knots (with a few exceptions) and lets us define a complexity index $c_J$ that will play a central role in the mass formula.
End-of-module quiz
Four questions to check your grasp of knot invariants.
1. What does Reidemeister's theorem say?
What is the difference between a chiral knot and an amphichiral knot?
3. For the trefoil knot $3_1$, $V_{3_1}(t) = -t^{-4} + t^{-3} + t^{-1}$. What is $V_{3_1}(1)$?
4. What is the index $c_J$ used for in TQNT?