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The Crossing Number

Beginner-Intermediate Transformation Unknotting number (crossing changes)
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Overview

A figure-eight knot wire frame has 4 crossings, each controlled by a removable steel pin. Flipping a pin swaps which strand goes over and which goes under at that crossing. The figure-eight knot has unknotting number 1 — a single crossing flip converts it to the unknot and frees the trapped ring. And on this diagram the situation is friendlier than the definition promises: every one of the four flips works. The puzzle is not a hunt for a magic crossing. It is a test of foresight: study the diagram, predict exactly which simplifying moves a flip will unlock, then flip and watch the frame dissolve on schedule.

Components

Part Material Dimensions
Figure-eight frame 4mm steel rod ~150mm across
Base Hardwood 150mm x 150mm x 20mm
Crossing pins (x4) 3mm steel rod 20mm tall, with 5mm ball caps
Pin sockets (x4) Drilled into frame 3.5mm diameter, at each crossing
Ring Welded steel O-ring 25mm OD, 3mm wire

Each pin sits in a socket at a crossing point. A groove in the pin holds the strands in their current over/under configuration. Removing and reinserting a pin upside-down swaps the crossing.

Setup

The Crossing Number - Setup Standard alternating figure-eight knot diagram made of steel rod, with 4 crossings at the north, east, south and west of the knot. Breaks in the strand show which strand passes under at each crossing, and the crossings alternate over-under along the wire. Each crossing has a removable pin, shown as a labeled circle (A, B, C, D) offset from the crossing and connected by a thin leader line. A ring is threaded on the upper-left arc of the wire. Wooden base below. Caption: one flip will unknot this; predict the moves it unlocks first. The Crossing Number — Setup figure-eight knot, 4 crossings, alternating A B C D ring One flip will unknot this. Predict the moves it unlocks first. figure-eight wire frame crossing pin (flippable) ring (trapped) gap = strand passes under
Figure-eight knot with 4 labeled crossing pins

  1. The figure-eight knot frame is mounted on the base
  2. Four crossing pins (labeled A, B, C, D) maintain the over/under pattern
  3. The ring is threaded onto the frame and can slide along the wire
  4. The ring cannot be removed while the frame forms a genuine figure-eight knot

Objective

Choose a crossing pin and, before touching it, predict the sequence of Reidemeister moves its flip will unlock. Then flip the pin and execute the simplification — trace the moves until the frame is an unknotted loop and the ring slides free. Flip first and you learn nothing: the puzzle is solved in your head before your hands move.

The Topology

The Figure-Eight Knot

The figure-eight knot (also called the Listing knot or knot 4_1) is the second-simplest non-trivial knot, with exactly 4 crossings in its minimal diagram. Unlike the trefoil, the figure-eight is amphichiral — it is equivalent to its mirror image. This makes it the simplest amphichiral knot.

What Is Unknotting Number?

The unknotting number u(K) of a knot K is the minimum number of crossing changes needed to convert K into the unknot. A crossing change swaps which strand goes over and which goes under at a single crossing.

Key values:

  • Unknot: u = 0 (it's already unknotted)
  • Trefoil: u = 1
  • Figure-eight: u = 1
  • 5_1 knot (cinquefoil): u = 2
  • Granny knot (trefoil # trefoil): u = 2

The figure-eight's unknotting number of 1 means that some single crossing change, in some diagram, converts it to the unknot. On the minimal 4-crossing diagram this puzzle is built from, the promise is over-delivered: every single flip works.

Every Flip Works on This Diagram

Crossing Options Table showing the result of flipping each crossing of the figure-eight knot's minimal diagram: every one of the four flips (A, B, C, D) yields the unknot and frees the ring. Flipping A or C opens the outer-shell clasp whose corners are crossings A and C; flipping B or D opens the central-eye clasp whose corners are B and D. Each route is a Reidemeister II slide that removes the opened clasp's two crossings, followed by two Reidemeister I untwists. Row C is highlighted as the worked example used in the text. Caption: on this diagram every flip works, so predict the route before you flip. Crossing Flip Options Flip Clasp it opens Route Result A outer shell (A–C) R-II, then R-I × 2 unknot ✓ B central eye (B–D) R-II, then R-I × 2 unknot ✓ C outer shell (A–C) R-II, then R-I × 2 unknot ✓ D central eye (B–D) R-II, then R-I × 2 unknot ✓ u(figure-eight) = 1 — every flip of this diagram realizes it On this diagram, every flip works — predict the route before you flip highlighted row = the worked example (C); circle = the unknotted frame, ring free flip A or C → the outer shell slides off; flip B or D → the central eye opens
All four crossing flips lead to the unknot, by two different Reidemeister routes

All four flips produce the unknot — what differs is the route the simplification takes:

Flip Clasp it opens Route to the unknot Result
A outer shell (corners A, C) R-II across the outer bigon, then two R-I untwists unknot — ring free
B central eye (corners B, D) R-II across the central bigon, then two R-I untwists unknot — ring free
C outer shell (corners A, C) R-II across the outer bigon, then two R-I untwists unknot — ring free
D central eye (corners B, D) R-II across the central bigon, then two R-I untwists unknot — ring free

The skill being tested is not choosing correctly — any choice is correct here — but seeing the dissolution before touching the cord: naming the clasp your flip opens and the moves that follow.

Why any crossing works here

The minimal figure-eight diagram is built from two clasps. Look at its faces: exactly two of them are bigons (two-sided regions). The outer shell is a bigon whose corners are crossings A and C — the two outermost arcs of the diagram cross each other there and nowhere else. The central eye is a bigon whose corners are B and D. Every crossing is a corner of exactly one bigon, so every crossing belongs to one clasp.

In the alternating diagram each clasp is locked: each of its two arcs goes over at one corner and under at the other, so neither arc can slide past the other. Flip either corner and the lock opens — one arc of that bigon is now over at both corners and the other under at both. That is precisely the configuration a Reidemeister II move removes: the over-arc slides off, and both crossings of the clasp vanish at once. The two surviving crossings, orphaned by the disappearance of their neighbors, degenerate into bare kinks — each undone by a Reidemeister I untwist. Four crossings, two clasps, every crossing a corner of one: every flip unknots the frame.

When the choice matters

The definition of u(K) minimizes over all diagrams of K and all choices of crossings — and that quantifier is where the difficulty of unknotting number lives:

  • This diagram's generosity is a fact about this diagram, not a law. On the cinquefoil (5_1, u = 2), no single flip of any diagram produces the unknot at all. On larger knots, different crossings of the same diagram genuinely lead to different knots, and choosing badly leaves you knotted.
  • No flip here — or anywhere — turns the figure-eight into a trefoil. On this diagram all four flips land on the unknot, and a 2012 theorem of Kawauchi shows that is no accident of the picture: no single crossing change in any diagram of the figure-eight yields a trefoil. The Gordian distance between the two knots is 2. Like unknotting number itself, distance between knots is a question about the knots, not about the diagram in front of you.
  • A knot where the choice genuinely decides the destination: 5_2. Its minimal diagram is a clasp plus a three-crossing twist region. Flip either clasp crossing and an R-II dissolves everything — the unknot. Flip a crossing in the twist region instead and two twists cancel, leaving a trefoil. Same diagram, different pins, different knots — the decisive-crossing hunt this puzzle spares you is waiting one knot up the table.

That is exactly why unknotting number is hard to compute: the optimal crossing change may not be visible in the diagram in front of you, and no simple algorithm tells you which diagram to look at.

Physical Intuition

What you feel in your hands: pull a pin out, flip it, push it back in. The steel never moves, yet the knot type changes instantly — the strand that was on top is now on the bottom. Slide the ring: where it used to jam against the opened clasp's crossings, it now finds slack, because the over-arc of that bigon no longer bars the way. The ring glides past every former catch point and off the end of nothing at all. The absence of catch points IS the unknot — and if you predicted the route, you knew exactly where the slack would appear before you felt it.

For the complete treatment of unknotting number and crossing changes, see Topology Primer: Unknotting Number.

Solution

The worked line below uses crossing C; the same method works from any pin.

  1. Examine. Find the diagram's two bigons: the outer shell (corners A and C) and the central eye (corners B and D). Note which clasp your chosen crossing belongs to — C is a corner of the outer shell.
  2. Predict. Flipping C puts the right-hand shell arc on top at both of its corners. Foresee the moves: an R-II slides the shell arcs apart (crossings A and C vanish), leaving B and D as bare single twists; two R-I untwists dissolve those, leaving a plain loop.

The Crossing Number - Checkpoint A: Predict The same alternating figure-eight diagram as the setup, annotated for the predict step. The outer face of the diagram is shaded: it is a bigon (a two-sided region) whose only corners are crossings A at the north and C at the south, where the two outermost arcs of the knot cross each other. Dashed circles mark those two corners. Pin C is highlighted green as the chosen flip, with the prediction written below: flipping C puts the right shell arc on top at both corners, so a Reidemeister II move slides the shell off, and the two remaining crossings B and D become single twists undone by Reidemeister I. The trapped ring still sits on the upper-left arc. The Crossing Number — Predict read the route off the diagram before flipping bigon corner bigon corner A B C D ring Predict: flipping C frees this bigon outer bigon — its only corners are A and C the clasp a flip of C (or A) unlocks chosen pin (C) — not yet flipped foreseen: R-II removes A & C, then B and D untwist (R-I)
Predict: flipping C frees the outer bigon — the shell will slide off by R-II

  1. Flip. Remove pin C, invert it, reinsert. The frame is now — topologically, before anything visibly moves — the unknot.
  2. Execute. Trace your predicted moves with the ring: slide it along the wire and it glides past both shell crossings (the R-II you foresaw), then past the two leftover twists (the R-I kinks). No catch points remain.

The Crossing Number - Checkpoint B: Execute Three panels showing the post-flip simplification, exactly as predicted. Panel 1: the figure-eight diagram with crossing C flipped (marked with a dashed green circle) - the right shell arc now rides over at both of the outer bigon's corners, so a Reidemeister II move slides the shell off, removing crossings A and C. Panel 2: the two surviving crossings B and D are now bare single twists, a loop with one small kink on each side, each undone by a Reidemeister I untwist. Panel 3: a plain unknotted loop; the ring is free beside it. Arrows labeled R-II and R-I x2 connect the panels. Caption: any pin ends the same way; flipping B or D opens the central eye first instead. The Crossing Number — Execute the flip unlocks the moves you predicted C flipped R-II R-I × 2 ring (free) 1. flip C: clasp opens 2. B, D are bare twists 3. nothing catches Execute the predicted route — each move was visible before the flip R-II slides the shell off, the two leftover kinks untwist, the ring falls free any pin ends the same way — flipping B or D opens the central eye first instead
Execute: the R-II slides the shell off, two R-I kinks untwist, ring free

  1. Free the ring. Slide it off the frame.

The Crossing Number - Solved The same alternating figure-eight diagram but crossing C (south) is flipped: the strand that passed under now passes over, drawn in green with a dashed highlight ring and a FLIPPED label. Pins A, B and D are gray; pin C is highlighted green. The diagram is now the unknot, so the ring has been freed: it sits at the upper right with a dashed exit-path arrow leading to it. C was the predicted flip worked in the text; on this minimal diagram any of the four flips unknots the frame. Label: Unknotting number = 1. The Crossing Number — Solved crossing C flipped, ring freed FLIPPED A B D C ring (free) exit path Unknotting number = 1 figure-eight wire frame (now unknotted) flipped crossing pin (C) ring (freed) exit path
Solved: crossing C flipped, ring freed

  1. Restore pin C and repeat from another pin if you like: A opens the same outer shell; B or D opens the central eye instead. Every route ends at the unknot — predicting which route is the whole game.

If your prediction was right, the dissolution feels anticlimactic. That is the point.

Why It's Tricky

The trap is behavioral, not structural. The puzzle looks like a search — four pins, one ring, surely one right answer — and that framing sends solvers into flip-and-test loops. But on this diagram every answer is "correct," so random flipping succeeds immediately and teaches nothing: the frame dissolves and the solver cannot say why.

The real challenge is to earn the flip: to look at a locked clasp and see, in advance, the R-II that isn't there yet. Solvers who do this discover something the flip-and-test solver never notices — that the four crossings pair into two clasps, and that a crossing change is not magic but simply the key that turns one clasp's lock.

Lesson

Unknotting number measures a knot's intrinsic distance from the unknot, minimized over all diagrams. This diagram makes u(4_1) = 1 visible four ways at once — and what it cannot show is just as instructive: no flip here produces a trefoil, and by a 2012 theorem of Kawauchi no flip in any other figure-eight diagram does either — the Gordian distance from figure-eight to trefoil is 2. Distances measured in crossing changes are invariants of the knots themselves, settled by no single picture of them. That is what makes crossing changes a fundamental — and genuinely difficult — operation in knot theory.

Common Mistakes

  1. Flipping before predicting. The flip will succeed — every flip here does — but you learn nothing. Restore the pin, and this time say out loud which bigon opens and which moves follow before you touch it.

  2. Predicting the wrong bigon. Each pin opens only its own clasp: flipping C opens the outer shell (A–C), not the central eye (B–D). If you foresee the wrong bigon, the moves you expect won't appear — the slack shows up at the other pair of crossings.

  3. Assuming this diagram's generosity generalizes. "Any crossing works" is a fact about the minimal figure-eight diagram, not about knots. On the cinquefoil no single flip works at all (u = 2), and on larger diagrams different flips lead to genuinely different knots.

  4. Confusing the wire's shape with the knot type. After the flip the frame still looks like a figure-eight — the steel hasn't moved. The knot type changed, not the geometry: the Reidemeister moves happen in your model (and in the ring's newfound freedom), not in the rod. Trust the ring, not your eyes.

Construction Notes

  • Bend the figure-eight frame from 4mm rod, ensuring crossings are clearly separated
  • At each crossing, one strand must pass 10mm above the other
  • Drill 3.5mm sockets at each of the 4 crossing points
  • Machine pins from 3mm rod: 20mm tall, with a 2mm groove at the midpoint that holds the strands
  • Ball caps (5mm) welded to pin tops for grip
  • The pin-and-groove mechanism must hold the strands firmly in the current over/under configuration
  • When flipped, the groove engages the strands in the reversed configuration
  • Mount the frame on a short post press-fit into the base
  • The ring (25mm OD) should slide freely along the wire when the knot is the unknot