Each knot leaves a fingerprint across a dozen invariant axes. Parallel coordinates reveal which invariants correlate, which are independent, and how the space of knots is shaped.
Two knots are equivalent if one can be continuously deformed into the other without cutting. There's no general algorithm to decide knot equivalence from diagrams, so the standard strategy is to compute quantities that stay fixed under deformation: knot invariants. Different invariant values prove inequivalence; matching values prove nothing.
Common invariants: crossing number, unknotting number, genus, bridge number, Alexander and Jones polynomials, determinant. Each captures a different aspect; none is complete — distinct knots can share every classical invariant.
Collecting these as a vector gives a point in a high-dimensional invariant space. Roughly 2,977 prime knots up to 12 crossings — point per knot, axis per invariant, polyline per row, the setting parallel coordinates is built for.
This article uses ten invariants per knot — nine non-negative integers plus the signed signature — taken from the Knot Atlas.
| Invariant | Symbol | What it counts | Unknot |
|---|---|---|---|
| Crossing number | c(K) | min crossings in any diagram | 0 |
| Genus | g(K) | min genus of Seifert surface | 0 |
| Unknotting number | u(K) | min crossing changes to unknot | 0 |
| Bridge number | b(K) | min local maxima in any projection | 1 |
| Braid index | br(K) | min strands in any braid word | 1 |
| Determinant | |Δ(−1)| | Alexander polynomial at t = −1 | 1 |
| Signature | σ(K) | signature of the Seifert form | 0 |
| Alex. poly. degree | deg Δ | span of Alexander polynomial | 0 |
| Jones poly. breadth | br(V) | max degree − min degree of V | 0 |
| Smooth 4-genus | g4(K) | min genus of smooth slice surface in B4 | 0 |
Table 1. Ten invariants used as axes throughout this article. The unknot takes the trivial value on every one. Any deviation from the "all-zero-except-bridge-and-braid-which-are-1" signature proves a knot is non-trivial, but two distinct knots can still produce the same row (see Figure 4).
Ten well-known prime knots: the unknot, the trefoil , the figure-eight , and the seven knots that together with the unknot fill out every prime knot with at most seven crossings. Each tile is a schematic diagram of the knot drawn from a parametric curve with over- and under-crossings indicated by breaks in the line.
Figure 1. Ten prime knots with crossing number at most seven, each drawn as a parametric closed curve. The knot names follow the Rolfsen table convention : the first number is the crossing number, the subscript enumerates distinct knots at that crossing number. The full Rolfsen table lists 1 knot at 0 crossings, 1 at 3, 1 at 4, 2 at 5, 3 at 6, 7 at 7, 21 at 8, 49 at 9, 165 at 10. These ten are only the beginning of the atlas.
Thirty-one knots from the Rolfsen table, each drawn as a polyline across ten invariant axes. Color runs from blue (low crossing number) through green to orange (high crossing number). Bundles (sets of polylines running nearly parallel) mean that a family of knots shares values across many invariants. Crossings between two adjacent axes mean the invariants are anti-correlated on that pair. Fans emerging from a single value on one axis indicate an invariant that does not discriminate: many knots pile into the same bin.
Figure 2. Thirty-one knots as polylines across ten invariant axes. The axes are ordered so topologically related invariants sit next to each other: crossings, genus, unknotting, bridge, braid (structural), then determinant and signature (Seifert-form), then the polynomial-span axes and smooth 4-genus. Even at 31 rows there is visible bundling: torus knots (, , , ) run along the high-genus, high-unknotting ridge while the twist knots pile up at low genus. The Kinoshita-Terasaka and Conway knots, the two eleven-crossing points at the top of the color ramp, dive to the bottom of the determinant and signature axes.
The axis order matters. Two invariants placed next to each other show their joint behavior; two invariants on opposite sides of the plot are visually uncoupled. Reordering is the main interactive gesture in parallel coordinates (see article 4). The order above puts the structural, numerical invariants first and the polynomial-derived ones last so you can compare them at a glance.
Bundles are clusters. The bundle you see on the left half of Figure 2 contains the unknotting-number-1 family: all knots you can undo with a single crossing change. Article 05 on bundles and deviations explains why that visual bundle is exactly what a high-dimensional clustering algorithm would find.
A ten-dimensional plot has pairs of coordinates. Between each pair there is a joint scatter that you can summarize with a single number: the Pearson correlation. Stack all 45 numbers into a 10×10 symmetric matrix with ones on the diagonal, and you get the correlation heatmap of the invariants. Dark red cells are strongly positive: when one invariant grows, the other grows. Dark blue cells are strongly negative. Cells near zero are near white: the two invariants essentially float independently across the sample of knots.
Hover any cell to read the correlation and see the invariants' names. The diagonal is always 1.
Figure 3. Pearson correlation matrix of the ten invariants computed over 31 knots. Red = positive, blue = negative, white = zero. The strongest positive correlations are the expected ones: crossing number with Jones breadth (~1.00, they nearly coincide for alternating knots), smooth 4-genus with unknotting number (~0.93), and genus with Alexander degree (~0.80). The weakest are determinant against unknotting, signature against crossings, and bridge against braid index, all near zero, evidence that these invariants carry independent information. The more blue or red the cell, the more redundant the axis pair.
A pair of invariants with correlation near 1 are almost redundant: you could drop one without losing much. A pair with correlation near 0 are nearly orthogonal: each one tells you something the other does not. The dimensionality of the invariant space is not really ten, but more like the number of independent directions in this correlation matrix. Principal component analysis on this matrix would show that the first two or three components already explain most of the variance, but the tails, the weakly correlated pairs, are where the most discriminating signal lives.
The unknot has a dull signature. Every invariant in Table 1 collapses to 0 or 1. That is the whole point of an invariant: a non-trivial knot should be visibly different. And for almost every knot in the atlas, the Alexander polynomial is different from the unknot's. Almost.
In 1957, Shinichi Kinoshita and Hidetaka Terasaka wrote down an eleven-crossing knot, now called the Kinoshita-Terasaka knot , that is demonstrably not the unknot, but whose Alexander polynomial is exactly . John Conway later found a second eleven-crossing knot, the Conway knot , related to by mutation (cutting a tangle out, rotating it 180°, and gluing it back). The two knots share not only the trivial Alexander polynomial, but also the same Jones polynomial. They are the most famous Alexander-polynomial conspirators in the tables.
On determinant (all 1) and Alexander degree (all 0), the three polylines agree; on those two axes alone, the Kinoshita-Terasaka and Conway knots are indistinguishable from the unknot.
Figure 4. Three conspirators against the Alexander polynomial. The unknot, Kinoshita-Terasaka (), and Conway () coincide on determinant and Alexander degree, both invariants computed from . On every other axis they separate. In particular, the Conway knot's smooth 4-genus is 1, while the Kinoshita-Terasaka knot is smoothly slice (). This is Lisa Piccirillo's 2020 result: the two knots are distinguished not by any classical polynomial but by the smooth slice genus.
Mutants share Alexander, Jones, HOMFLY polynomials, signature, and hyperbolic volume. Piccirillo's 2020 proof that the Conway knot is not smoothly slice used a different knot with computable 4-genus, transferring the result via concordance. With 10 axes, two distinct knots can overlap on 8 — adding invariants finds where they diverge.
The Alexander polynomial is a Laurent polynomial in one variable with integer coefficients. For the knots in our table, it rarely has more than 7 terms. Instead of collapsing it to a single number (the degree, or the determinant), we can use every coefficient as its own axis. The trefoil has , so it becomes the row . The figure-eight has , giving . Stacked across all knots, this produces a much wider parallel-coordinates view: one polyline per knot, one axis per coefficient.
The plot below shows all 31 knots with their Alexander polynomials expanded to degree-3 axes ( through : seven axes, covering every coefficient of the polynomials in our sample). Because the Alexander polynomial is symmetric (), the plot is mirror-symmetric around the middle axis when read from outer axes inward. That symmetry is visible as a bilateral pattern in the polylines.
Figure 5. Alexander polynomial coefficients as parallel-coordinates axes. The axis labels are the powers of . Torus knots (, , , ) form a visible ridge that oscillates between positive and negative coefficients. Twist knots bunch near zero on the outer axes and spike at the center. The Kinoshita-Terasaka and Conway knots collapse to the identity polyline (all coefficients zero except ), which is the same as the unknot.
Invariants do not float freely. There are classical inequalities between them that act as lower and upper bounds, and those bounds show up in parallel coordinates as ordering constraints: if invariant is always at most invariant , then on adjacent axes no polyline can jump upward from the -axis to the -axis by more than the gap allows, and more importantly, on a normalized scale every polyline must be monotone non-increasing or non-decreasing in that pair. Three classical bounds appear in our data:
The first is the genus bound on the unknotting number. The second is Bennequin's inequality relating crossings and genus. The third is Milnor's inequality relating smooth 4-genus to unknotting number. Figure 6 overlays these bounds as shaded envelope regions against the polylines that satisfy them. Below each pair of axes, the readout reports how many of our knots saturate the inequality (achieve equality) and how many sit strictly inside.
Figure 6. Knot invariant bounds visualized as envelopes between adjacent parallel-coordinates axes. The smooth 4-genus is at most the unknotting number (each crossing change cobounds a disk in ) and at most the 3-genus (push the slice surface to the boundary). The Alexander polynomial degree is bounded above by twice the 3-genus. And for every knot in our sample, twice the genus is at most the crossing number. These four inequalities shape the "orthant" where valid knot invariant vectors live; not every row of non-negative integers is realized.
The ten invariants aren't independent. Genus, bridge, unknotting, and 4-genus sit in a tight web of inequalities; braid index, crossing number, and Seifert form are coupled. The space of knots is a thin curved subset of ℤ¹⁰; the polyline bundle outlines the constraint envelopes.
This is the same pattern as the outlier signature in article 5: "outside the bundle" would mean violating an inequality, which can't happen for a real knot.
Parallel coordinates turn classical knot tables into a readable atlas. The same approach handles any catalog characterized by invariants: exceptional root systems, finite groups with Cayley graph diameters, polytopes with face numbers and symmetry orders.
The next wave of invariants — Khovanov homology, Heegaard Floer, the Witten-Reshetikhin-Turaev quantum invariants — adds axes that catch distinctions classical invariants miss. The Conway and Kinoshita-Terasaka knots, indistinguishable pre-1980, separate once smooth 4-genus is added.