Topological quantum field theory · Jones invariants

Knots & Quantum Fields · Witten’s Bridge

Drag the projection, continuously stretch and shear a closed knot, and inspect the three Reidemeister moves. A planar diagram can gain or lose crossings, yet the knot type and Jones polynomial remain fixed unless the strand is cut or passed through itself. Then reveal conceptual field histories to see how a Wilson loop places the knot inside Chern–Simons quantum field theory.

Knot typeTrefoil

Visible diagram crossings $N_{\times}=3$

Minimum crossing number $c(K)=3$

Jones fingerprint · fixed convention $V_{3_1}(q)=q^{-1}+q^{-3}-q^{-4}$

Chern–Simons sampling point $q=e^{2\pi i/(k+2)}$

Current allowed moveI · twist / untwist

The field lines conceptually represent a sum over gauge-field configurations; they are not a numerical path-integral solution. Known reference values show that $V_K(q)$ stays fixed under continuous deformation. Its precise Chern–Simons identification also depends on orientation, mirroring, normalization, and framing conventions.

Physics tutorial

From a Closed String to Quantum Field Theory

BackgroundA mathematical knot is not a shoelace with two ends. It is a non-self-intersecting closed curve in three-dimensional space. Two shapes represent the same knot when one can be deformed into the other without cutting, gluing, or passing strands through each other. Their planar diagrams may change, and the three Reidemeister moves exactly encode that diagrammatic equivalence.

Why it mattersThe Jones polynomial was first discovered by mathematical methods. In $1989$ , Witten showed that a knot used as a Wilson loop in $2+1$ -dimensional Chern–Simons quantum field theory produces Jones-type knot invariants. Physics did not guess an answer for mathematics; it revealed a natural three-dimensional quantum theory behind apparently separate algebraic rules.

Start with the essentials

Focus question: A planar knot diagram can gain or lose crossings. Why can a quantum-field calculation depend only on the knot type and not on the chosen drawing?
One-sentence intuition: The Chern–Simons action needs no spatial metric, while a Wilson loop records gauge-field parallel transport around a closed curve. Their quantum expectation value is therefore insensitive to smooth coordinate deformations—the exact behavior required of a knot invariant.

Core mathematical model

A knot deformation stays embedded

K_s:S^1\hookrightarrow \mathbb R^3,\qquad 0\le s\le1

Every $K_s$ must remain a non-self-intersecting closed curve. Stretching, rotating, and shearing are allowed. If two strands pass through each other at any instant, the map stops being an embedding and the knot type may change.

A Wilson loop inserts the knot into field theory

W_R(K)=\operatorname{Tr}_R\,\mathcal P\exp\!\left(\oint_K A\right)

$A$ is a gauge connection and $\mathcal P$ orders it along the path. One can picture a probe with an internal quantum “color” traveling once around $K$ and recording the total parallel transport caused by the gauge field.

The Chern–Simons action uses no metric

S_{\mathrm{CS}}[A]=\frac{1}{4\pi}\int_M\operatorname{Tr}\!\left(A\wedge \mathrm dA+\frac{2}{3}A\wedge A\wedge A\right)

The expression uses orientation, differential forms, and wedge products on a three-manifold, but no lengths, angles, or distances. This metric-free structure is the central reason its quantum observables can become topological invariants.

The quantum expectation meets the Jones polynomial

\left\langle W_R(K)\right\rangle=\frac{1}{Z}\int\mathcal DA\,e^{ikS_{\mathrm{CS}}[A]}W_R(K)\ \longleftrightarrow\ V_K(q),\qquad q=e^{2\pi i/(k+2)}

For $SU(2)$ , the fundamental representation, and suitable normalization and framing conventions, the Wilson-loop expectation gives the Jones polynomial at roots of unity. The field histories at right explain this logic; the browser is not evaluating the infinite-dimensional path integral.

Common difficulties

Witten did not invent the Jones polynomial from nothing

Typical misconceptionMathematicians became stuck, so Witten wrote a set of physics formulas no one else dared to imagine and directly created the invariant.

Better mental modelVaughan Jones had already discovered the polynomial, and algebraic and statistical-mechanical constructions already existed. Witten’s breakthrough was a natural three-dimensional quantum-field interpretation that unified knots, polynomials, gauge fields, and topological quantum field theory.

The path integral does not average knot shapes

Typical misconceptionQuantum field theory averages every distorted drawing of one knot, so ordinary shape disappears.

Better mental modelThe path integral keeps the manifold and Wilson loop $K$ fixed while integrating over gauge-field configurations $A$ . Topological invariance comes from the structure of the action and observable, not from an ordinary average of knot diagrams.

Diagram crossing count is not a knot invariant

Typical misconceptionA three-crossing drawing must show exactly three crossings from every viewpoint.

Better mental modelChanging the projection or applying Reidemeister moves can alter the displayed crossing count. The minimum crossing number $c(K)$ is minimized over all diagrams, while the Jones polynomial remains unchanged because its local relations are compatible with all three moves.

The Jones polynomial is not a complete identity card

Typical misconceptionTwo knots with equal Jones polynomials must be the same knot.

Better mental modelDistinct knots can share one Jones polynomial. It is a powerful invariant that separates many knots, but it is not a complete classifier.

Run the experiment

01
Begin with continuous deformation only
Choose the trefoil, start the deformation, then drag the canvas to change its projection.
What to observe: The outline and visible crossing count can change, yet the knot type, minimum crossing number, and $V_K(q)$ remain fixed because no strand passes through another.
02
Inspect the three local moves
Press “Next Reidemeister move” repeatedly and compare the before-and-after diagram at lower right.
What to observe: Move I changes a local curl, move II adds or removes two crossings, and move III slides one strand past a crossing of two others. Each changes the drawing without changing the three-dimensional knot.
03
Compare three known knot types
Select the unknot, trefoil, and figure-eight in turn. Change the projection and deformation for each.
What to observe: The presets have different reference polynomials, so $V_K(q)$ separates these three knots. Every allowed deformation inside one preset retains the same fingerprint.
04
Cross Witten’s bridge
Reveal the conceptual gauge-field histories, change the Chern–Simons level, and read the three steps at right.
What to observe: The knot first becomes a Wilson-loop observable, then receives a quantum average over gauge fields. Changing the integer level $k$ changes the root of unity $q=e^{2\pi i/(k+2)}$ —a different quantum sample of the same Jones polynomial.