The Jones polynomial skein relation determines $V_K$

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.