Categorification replaces algebraic structures with richer categorical ones: numbers become vector spaces (dimension recovers the number), polynomials become chain complexes (Euler characteristic recovers the polynomial), and equalities become isomorphisms. For knot invariants: if $P(K)$ is a polynomial knot invariant, a categorification is a bigraded (co)homology theory $H^{i,j}(K)$ such that $P(K) = \sum_{i,j}(-1)^i q^j \dim H^{i,j}(K)$. The homology groups carry strictly more information than the polynomial.

Given a knot diagram $D$ with $n$ crossings, construct the Khovanov chain complex:

1. Cube of resolutions: Each crossing can be resolved in two ways (0-resolution: horizontal, 1-resolution: vertical). A binary string $\alpha \in \{0,1\}^n$ specifies a complete resolution $D_\alpha$ -- a collection of disjoint circles.

2. State spaces: To each resolution $D_\alpha$ with $k$ circles, assign $V_\alpha = V^{\otimes k}$ where $V = \mathbb{Q}\{v_+, v_-\}$ is a 2-dimensional graded vector space ($\deg v_+ = 1$, $\deg v_- = -1$). The chain group is $C^i(D) = \bigoplus_{|\alpha|=i}V_\alpha$ where $|\alpha| = \sum\alpha_j$.

3. Differentials: For $\alpha, \beta$ differing in one coordinate (a single crossing change), define $d: V_\alpha \to V_\beta$ using the merge map $m: V\otimes V\to V$ (when two circles merge: $v_+\otimes v_+ \mapsto v_+$, $v_+\otimes v_- = v_-\otimes v_+ \mapsto v_-$, $v_-\otimes v_- \mapsto 0$) or the split map $\Delta: V\to V\otimes V$ (when one circle splits: $v_+ \mapsto v_+\otimes v_- + v_-\otimes v_+$, $v_- \mapsto v_-\otimes v_-$). Signs are assigned to make $d^2 = 0$.

4. Khovanov homology: $\mathrm{Kh}^{i,j}(K) = H^i(C^{\bullet,j}(D))$ (after grading shifts depending on the number of positive and negative crossings).

Khovanov homology satisfies: (1) Knot invariance: $\mathrm{Kh}(K)$ is invariant under Reidemeister moves (proved by verifying chain homotopy equivalence). (2) Functoriality: a cobordism $\Sigma: K_1 \to K_2$ (a surface in $S^3\times[0,1]$ bounded by $K_1$ and $K_2$) induces a map $\mathrm{Kh}(\Sigma): \mathrm{Kh}(K_1) \to \mathrm{Kh}(K_2)$. (3) Unknot detection: $\mathrm{Kh}(K) \cong \mathrm{Kh}(U)$ implies $K = U$ (Kronheimer-Mrowka, 2011, using gauge theory). This is stronger than the Jones polynomial, which is not known to detect the unknot.