Knot theory

This gives the first rigorous proof that a knot can be genuinely knotted — not just hard to untangle by hand, but provably inequivalent to the unknot — using only elementary combinatorics, no advanced machinery.

Invariance under Type I. A Type I move adds or removes a small loop, creating one crossing where a single strand crosses itself. In any valid tricoloring before the move, the strand has one color, say $a$, entering and leaving the loop. Adding the twist creates a crossing with under-strands both colored $a$ and over-strand also forced to color $a$ (since the loop is a single strand), and $a+a+a=3a\equiv 0\pmod 3$ always holds automatically — so the coloring extends validly to the new crossing without constraint, and conversely restricts validly when removing it. The overall colorability (existence of a valid coloring using $\ge 2$ colors) is preserved.

Invariance under Type II. A Type II move introduces or removes two crossings where two strands, colored $a$ and $b$ (possibly equal), cross twice. Checking both crossings: at each, the two strands present are $a,b$ and the third (the over-strand at each crossing, which is one of the same two strands continuing through) is forced by the rule to be whichever color makes $a+b+c\equiv0$; since the two crossings are related by the strands simply passing over each other, the coloring on the two original strands $a,b$ extends consistently to color both new crossing regions without introducing new colors or contradictions, and removing the move just deletes those two constraints, which were automatically satisfiable. So colorability is preserved both ways.

Invariance under Type III. A Type III move slides a strand across a crossing, rearranging three crossings among three strands colored $a,b,c$ (say) without changing which colors appear where globally — it only changes the local combinatorial arrangement of the same three colored arcs. Since the coloring rule $x+y+z\equiv0\pmod3$ at each of the three crossings depends only on the colors of the (unordered) triple of strands meeting there, and Type III does not change which strands meet at crossings (just their diagram-local arrangement), a valid coloring before the move restricts to a valid coloring after, and vice versa.

Conclusion. Since tricolorability (as a yes/no property, requiring $\ge 2$ colors used) is unchanged by all three moves, it is an invariant of the knot, not just the diagram. The standard trefoil diagram admits the coloring $a=0,b=1,c=2$ on its three arcs (one checks all three crossings satisfy $0+1+2=3\equiv0\pmod3$), using $3\ge2$ colors, so the trefoil is tricolorable. The standard unknot diagram (a single unknotted loop, no crossings, or any diagram reducible to one arc) has only one arc, hence only one color is available, failing the "$\ge 2$ colors" requirement — the unknot is not tricolorable. Since tricolorability is an invariant and the two diagrams disagree, no sequence of Reidemeister moves connects the trefoil diagram to the unknot diagram, so the trefoil is a genuinely knotted circle.

The skein relation turns an intractable-looking 3D deformation problem into pure recursive algebra: any knot diagram can be reduced to the unknot by repeatedly resolving crossings, and the relation tells you exactly how the polynomial changes at each resolution, making $V_K(t)$ algorithmically computable.

Well-definedness sketch via crossing induction. Order the crossings of a diagram $D$ and define $c(D)$ as the number of crossings that must be switched to reach a diagram of the unknot (this is always finite, since switching all crossings of any diagram in a suitable order unknots it — a classical fact). We show $V_K(t)$ can be computed by induction on $c(D)$.

Base case. If $c(D)=0$, $D$ already represents the unknot (possibly with extra disjoint unknotted circles from Reidemeister-I-type simplification), and $V_K(t)=1$ by definition (with a normalization factor $(-t^{1/2}-t^{-1/2})^{k-1}$ for $k$ disjoint unknotted components, consistent with the skein relation applied to split unlinks).

Inductive step. Suppose $c(D)\ge 1$, and pick a crossing whose switch reduces $c$. Let $D_+$ (or $D_-$) be the diagram at that crossing before switching and $D_-$ (or $D_+$) after switching — by construction $c$ decreases for one of them. Let $D_0$ be the same diagram with that crossing smoothed (removing it, connecting strands the other way); $D_0$ has one fewer crossing overall so also has smaller $c$ (or is handled by the base case). By the inductive hypothesis, $V$ is already known for $D_-$ (or $D_+$) and $D_0$. Rearranging the skein relation $t^{-1} V_{L_+} - t V_{L_-} = (t^{1/2}-t^{-1/2})V_{L_0}$ to solve for the unknown term (e.g. $V_{L_+} = t\big(V_{L_-} + (t^{1/2}-t^{-1/2})V_{L_0}\big)$) computes $V$ for $D$.

Consistency (sketch). The nontrivial part of the theorem — that this recursively computed value does not depend on the order of crossing choices, and is invariant under all three Reidemeister moves — is established by Kauffman's bracket polynomial construction: define $\langle D\rangle$ by local smoothing rules $\langle\text{crossing}\rangle = A\langle\text{smoothing }0\rangle + A^{-1}\langle\text{smoothing }\infty\rangle$ and $\langle D\sqcup\bigcirc\rangle = (-A^2-A^{-2})\langle D\rangle$; one checks directly (a finite local computation) that $\langle D\rangle$ is invariant under Reidemeister II and III, and changes by a controlled factor under Reidemeister I, which is exactly compensated by multiplying by $(-A^3)^{-w(D)}$ where $w(D)$ is the writhe (signed crossing count); substituting $t=A^{-4}$ recovers $V_K(t)$ satisfying exactly the stated skein relation.