An Introduction to Knots

What a knot is — and what it is not

Pick up a loop of string, tie an ordinary overhand knot in it, then fuse the two ends together so there is no way to slip the knot off the end. That sealed, tangled loop is a knot in the mathematical sense: a circle sitting inside ordinary three-dimensional space, possibly tangled with itself. The sealing of the ends is the whole point. As long as a string has loose ends you can always untie any knot by threading an end back through, so an open string is topologically boring; only the closed loop can trap a genuine tangle. A knot, then, is a circle — the simplest closed curve there is — but the interesting information is not in the circle, it is in how the circle is placed in space.

This is a subtle and important shift, so let us be honest about it. Considered all by itself, every knot is the same space: any knotted loop, no matter how snarled, is just a circle, and is homeomorphic to the round unknotted circle, because intrinsically a circle is a circle. The knottedness is therefore not a property the loop has on its own; it is a property of the embedding — the way the loop is threaded through the surrounding space, together with what that space won't let you do. Knot theory studies circles up to a stricter sameness than homeomorphism: two knots count as the same knot only when you can slide one to the other through space, dragging the string around continuously without ever letting it pass through itself.

Drawing a knot: diagrams and crossings

We cannot do mathematics with a tangle of real string, so the first move is to flatten it onto paper. Cast a shadow of the knot onto a plane and you get a knot diagram: a closed curve that crosses itself at a handful of points, with a tiny break drawn at each crossing to record which strand passes over and which passes under. That over/under bookkeeping is the only thing the flat picture must not lose — it is the whole memory of the third dimension we threw away by flattening. Wipe out the over/under data and you cannot tell a knot from its mirror image, or even from the unknot.

The simplest genuinely knotted diagram is the trefoil: a loop that crosses itself three times in a tidy three-fold pattern, like a pretzel with one extra twist. The plain circle drawn with no crossings is the unknot — the loop with no knot in it at all. In between, a single loop with one or two crossings can always be wiggled flat back to the unknot; you cannot trap a true knot with fewer than three crossings. The trefoil's three crossings are the minimum price of being knotted, which is why it is the first knot in every table and the worked example we return to throughout this guide.

      ___              The TREFOIL needs 3 crossings.
     /   \             At each crossing one strand is
    |     |            drawn broken (it passes UNDER);
  --+--.  |            the unbroken strand passes OVER.
    |  '--+--
    |     |            0 crossings  ->  the UNKNOT
     \___/             1 or 2       ->  still the unknot
                       3 crossings  ->  first real knot

The three Reidemeister moves

Here is the difficulty laid bare. One knot has infinitely many diagrams, because you can always jiggle the string and redraw it with more crossings or fewer. So when two diagrams look different, you face the same awful asymmetry we met with homeomorphism: showing they are the same knot is easy once you find the right sequence of wiggles, but showing they are different seems to demand checking infinitely many redrawings. The breakthrough is a theorem of Reidemeister: any two diagrams of the same knot can be connected by a finite sequence of just three local moves, plus ordinary sliding of the strands that changes no crossings.

1. Move I (twist): put a single kink into a strand, or pull one out — adding or removing one self-crossing where the string loops over itself once.
2. Move II (poke): slide one strand over another so two crossings appear together, or pull it back so two crossings cancel.
3. Move III (slide): slide a strand across the point where two other strands cross, leaving the count of crossings the same but rearranging which lies near which.

Reidemeister's theorem changes the whole game. It turns a question about continuous motion in 3D — drag the real string around without self-passage — into a finite, combinatorial question about flat pictures: can you get from this diagram to that one using only Moves I, II, III? This is the same liberation the rest of the rung kept handing us: replace 'check infinitely many deformations' with 'check the answer is unchanged by a few generating moves'. The recipe for a knot invariant now writes itself: build any quantity from a diagram, then prove it does not change under each of the three moves.

Three-colouring: a proof the trefoil is knotted

Let us actually build one such invariant and watch it pin down the trefoil. The rule is tricolourability. Take the diagram and look at its arcs — the unbroken pieces of curve that run from one under-crossing to the next. Try to paint every arc one of three colours so that two conditions hold: at least two different colours appear somewhere, and at every crossing the three arcs meeting there are either all the same colour or all three different. A diagram that admits such a painting is called tricolourable.

Now test the two knots by hand. The unknot's plain circle is a single arc; with only one arc you can use only one colour, so the 'at least two colours' rule fails — the unknot is not tricolourable. The trefoil has exactly three arcs, and if you give each arc its own colour, then every one of the three crossings sees all three colours at once, which the rule allows — the trefoil is tricolourable. The two knots disagree on this property, so they cannot be the same knot. The trefoil genuinely cannot be untied: it is the honest, finite proof that knots exist, and it never once mentions the third dimension.

From loops to knots, and the honest frontier

Tricolourability is the gentlest of many knot invariants, and it is not perfect: it cannot tell every pair of distinct knots apart, and it cannot, by itself, distinguish the trefoil from its own mirror image — yet those two trefoils really are different knots, a fact that needs a sharper tool. The deeper invariants reach back to the previous guide. Instead of studying the knotted loop itself, study the space left over when you delete the knot from three-dimensional space — the knot complement — and take its fundamental group, watching how loops in the surrounding air can or cannot be reeled past the missing string.

This is exactly the homotopy of loops you just learned, now asked of a punctured space, and it is astonishingly powerful: the fundamental group of the complement is strong enough to tell the unknot from every nontrivial knot, and the trefoil from its mirror. The honest catch is that computing and comparing these groups is real, demanding algebra — well past what an introductory rung can carry out. We can faithfully describe what the invariant is and why it works without pretending the calculation is a one-liner; the full machinery genuinely belongs to a later course.

Step back and see the shape of the whole rung. You began with homeomorphism and the rubber-sheet view, learned to count holes with the Euler characteristic, sorted surfaces by genus, and caught holes with loops via the fundamental group. Knots are where all of it converges: a knot is a tame circle, told apart not by anything intrinsic but by how it sits in space, and the way you tell two of them apart is, once again, by finding an invariant on which they disagree. Same strategy, sharper stage — and an open frontier, since classifying all knots completely is, to this day, an active and genuinely hard problem.