The Name and the Knot
In 1974 Kenneth Perko showed that two entries in Rolfsen's ten-crossing knot table were the same knot, drawn twice. Fifty years later the mix-up is still live: SnapPy's shipped Rolfsen table calls the surviving knot both 10_161 and 10_162, so its 10_162 and the Knot Atlas's 10_162 are different knots. Ask four sources of the same table, then count the pairs of distinct knots each old certificate could never tell apart.
The check
Every number this page prints comes from research/perko-pair/verify.mjs, which recomputes the Jones and Alexander polynomials of every prime knot with at most ten crossings from scratch, from three descriptions of each diagram (the Knot Atlas's PD code, its braid word, and a rebuild of its DT code by spherogram), requires them to agree, and checks them against the ones KnotInfo publishes: .../... checks passed.
The knot Perko drew twice
Rolfsen's Appendix C, page 415 of the 2003 corrected printing, still lists a 10_162. Its Conway symbol is 21:-20:-20, its drawing is the second image below, and beneath it Rolfsen prints one line: "(K. Perko has pointed out that 10162 is actually equivalent to 10161.)" Every ten-crossing knot the corrected table then draws already had a determinant, printed alongside as a bracketed shorthand. So we have Rolfsen's own writing of the answer, and we can put a number on it.
The two diagrams the Wasteland's verifier reads on the left both draw the knot the modern tables call 10_161: they have the same Jones polynomial up to mirroring, the same Alexander polynomial, and the same hyperbolic volume, 5.6387729. The third diagram is a different knot: it is the one that the Knot Atlas, KnotInfo and Wikipedia now call 10_162, which is Rolfsen's 10_163 after Perko's duplicate was pulled out and the tail slid down by one.
Four tables, one page from the book
Rolfsen's own final page of the ten-crossing table is reproduced from the scan, and the bracketed Alexander polynomials are read out (his shorthand: [a-b+c-d+e stands for a + b(x+x^-1) + c(x^2+x^-2) + ..., so the determinant is that expression at x = -1). What follows in the last two columns is not asserted: it is the determinant three modern software packages return when you ask them for the knot they call by the same name.
| Rolfsen p. 415 | Conway | printed | = det | SnapPy | Atlas / KnotInfo (same name) |
|---|
SnapPy's column reproduces Rolfsen's determinant on every row of page 415. The Atlas/KnotInfo column, keeping the SAME name, does not: it diverges at 10_162, where those tables put a different knot under the name because they closed the gap and slid every later name down.
Where the numbering goes bad, name by name
Only seven ten-crossing names denote different knots in SnapPy than in the Knot Atlas and KnotInfo. Five are the tail Perko exposed; two are an older 1976 mix-up of 10_83 and 10_86, where the Conway symbols and Alexander polynomials in Rolfsen's first edition were attached to each other's diagrams. Rolfsen's 2003 corrected printing swapped the invariants, not the diagrams; SnapPy is loading from a triangulation table whose names were pinned in the 1976 sense. So the disagreement is not that anyone made an arithmetic error, and neither table is wrong; the arithmetic is just that a NAME denoting a knot only up to mirror image, and only up to which numbering convention someone is loading, is a lossy label. It has to be re-cast every time the software boundary is crossed.
| the same name... | SnapPy 3.3.2 | Knot Atlas / KnotInfo | Hoste-Thistlethwaite | reason |
|---|
Given a name, which knot?
Pick a ten-crossing name and see what the two tables mean by it, drawn from the crosswalk this study built. The one row where SnapPy has a name the other tables no longer publish is 10_166, which is Rolfsen's; the two tables that dropped the duplicate stop at 10_165. That one label alone is enough to show the shift, but the interesting rows are the middle five.
What the tabulator ever had to work with
A tabulator can prove that two entries are DIFFERENT. There is no obligation to prove they are the same (that is what a duplicate is), so a table is only ever as safe as the invariants it can compute for every entry, and no safer than the pair of entries the invariants happen not to separate. Below is the ladder of certificates a tabulator can reach, and the count of pairs of GENUINELY DISTINCT knots each certificate fails to separate. The count is over all 30,876 pairs of distinct knots in the modern table.
The determinant fails to separate 542 pairs; the whole classical arsenal by 1974 (the Alexander polynomial, and Trotter and Murasugi's signature bought a decade earlier) fails to separate 31. The Jones polynomial arrived in 1984, ten years too late to help Perko, and even together with the Alexander polynomial it leaves five undecided pairs. Thurston's volume of the hyperbolic complement is the first single certificate that leaves only one pair, the pair 9_42 and 10_132, and it is undefined for the six torus knots. The isometry signature of the complement, a canonical string for the hyperbolic structure, separates every pair it is defined on.
A pair of ten-crossing knots the Jones polynomial cannot see: it fails on 10_25 and 10_56 (which the Alexander polynomial catches) and on the sisters 10_40 and 10_103 (which it does not, so they need the signature or the volume to be told apart). And a pair that no polynomial in this list can catch: 5_1 and 10_132 agree on both the Alexander and the Jones polynomial, but they are not the same knot; the volume of 5_1 is undefined (it is a torus knot) and the volume of 10_132 is 4.05686022, so the geometry pulls them apart the moment the volume is asked.
The one pair no single invariant on this ladder can separate
The knots 9_42 and 10_132 are the strangest row in the whole ladder. They have distinct Alexander polynomials and distinct Jones polynomials and distinct signatures, so no polynomial certificate ever misses that they are different knots. But SnapPy computes their hyperbolic volume to be the same 4.05686022... to eleven decimal places at ordinary precision (they are commensurable, sharing a covering manifold); at 63 decimal places their volumes are still the same. The one invariant on the ladder that separates them for good is the isometry signature of the complement, which is a canonical string for the underlying hyperbolic structure and returns different strings for the two. A volume ties, a canonical structure does not.
The count no one has ever printed
The table as it stood before 1974 had 250 entries, with two of them the same knot. In that table, how many pairs of entries share an Alexander polynomial? 41. Jones? 15. Both? 6. In each case exactly one MORE than the count in the modern table (40, 14, 5), and the extra pair in every case is exactly the pair Perko caught: 10_161 and the deleted 10_162 lived inside the smallest coincidence class in the whole table, the two-knot class the Jones polynomial cannot pull apart from itself. The hardest place to spot the duplicate in the entire body of pre-1974 tools is where it actually was.
How the recomputation is done, in plain terms. For every knot the Knot Atlas ships (there are 249, running 3_1 through 10_165), the immersive verifier here reads the planar-diagram code the Atlas publishes, walks its Kauffman bracket state sum over all 2n resolutions, normalises by the writhe, and reports the Jones polynomial. It walks the Wirtinger presentation and Fox-differentiates it, then takes an exact BigInt determinant of a symbolic matrix (via a small Lagrange interpolation over the rationals), and reports the Alexander polynomial. Both are cross-checked against the same computation on the BRAID WORD the Atlas publishes, and against a spherogram rebuild of its DT code, and against the polynomials KnotInfo publishes as prose. The recomputed determinant equals the Atlas's for every one of the 248 knots it publishes, and equals KnotInfo's for every one of the 249. All 24 checks green.
Sources, and what is not asserted
Primary source. Dale Rolfsen, Knots and Links, Publish or Perish 1976; reprinted with corrections by the American Mathematical Society, 2003 (AMS Chelsea 346.H). Appendix C, "Table of Knots and Links". Read from the scan at Andrew Ranicki's archive, University of Edinburgh: webhomes.maths.ed.ac.uk/~v1ranick/papers/rolfsen.pdf. The two pages this study depends on are transcribed by hand into research/perko-pair/data/rolfsen-appendix-c.json and used only as data, never as narration.
Primary source. Kenneth A. Perko, Jr., "On the classification of knots", Proceedings of the American Mathematical Society 45 (1974), 262-266. AMS PDF. Perko's own writing labels the pair in Little's 1900 notation (105II and 106VI), not Rolfsen's; the Rolfsen labels are the ones that stuck.
Software tables. The Knot Atlas at katlas.org, pages fetched once on 2026-08-20 into research/perko-pair/data/katlas/. KnotInfo at knotinfo.org, read through the database_knotinfo package (version 2026.8.1) that SageMath itself uses. SnapPy version 3.3.2 shipped Rolfsen table, via the snappy.LinkExteriors census; the triangulations are attributed to Joe Christy.
Not asserted. Rolfsen prints a determinant of 5 under the deleted 10_162. That is because the invariants he prints under the two entries agree, which is what Perko's proof reduces to at the level of the printed table: they are the same knot, so their invariants have to match. This study does not claim to reproduce the deleted diagram from Rolfsen's 1976 first edition, only its determinant (which SnapPy still ships). The Kawauchi textbook takes an opposite convention (it exchanges DIAGRAMS to fix the 1976 mix-up of 10_83 and 10_86, rather than exchanging invariants); this study does not adjudicate that. Nothing here rests on which mirror image a table draws; the crosswalk matches knots on the Alexander polynomial, the Jones polynomial up to mirroring, and the isometry signature of the complement where SnapPy can compute one.
Reproducibility. One command, offline once the Knot Atlas pages are cached and the two Python probes are run: node research/perko-pair/verify.mjs. It exits 0 on 24/24 green and 1 on any failure. Every number this page prints reads from public/strata/the-name-and-the-knot/data.json, which is written by research/perko-pair/page-data.mjs, which reads only the artifacts the verifier writes.