Parity-of-Crossings Theorem

• Parity-of-Crossings Theorem is a unifying principle that assigns abelian group values to crossings in knot diagrams, ensuring that local Reidemeister rules determine global parity outcomes.
• It underpins the construction of knot and link invariants by linking local crossing labels to global topological features in contexts such as virtual knots, amplituhedra, and Lorentzian geometry.
• By employing inductive proofs based on Reidemeister moves, the theorem provides a robust framework for defining parity functors and refining invariants including parity-graded homology.

The Parity-of-Crossings Theorem provides a unifying structural rule across knot theory, algebraic combinatorics, and Lorentzian geometry: the assignment of group-valued (often abelian) labels to crossings, subject to specific local relations, implies global parity laws on the sums or products of crossing values around certain topological or combinatorial features ("polygons," loops, or cycles). This theorem is foundational in the theory of parity functors for knots, the construction and invariance of certain knot and link invariants, the combinatorics of positroidal cell decompositions such as amplituhedra, and the global topology of spacetimes with nontrivial covering spaces.

1. Definitions and Fundamental Structures

The Parity-of-Crossings Theorem formalizes conditions on labels assigned to diagrammatic crossings in various contexts. For knot diagrams (virtual, flat, free, or in surfaces), a parity is a rule $p_D : V(D) \to A$ assigning values in an abelian group $A$ to the set $V(D)$ of crossings in a diagram $D$, subject to axioms matched to the three types of Reidemeister moves:

• (P0) Invariance: surviving crossings under moves retain their parity,
• (P1) First move: a disappearing R₁ crossing must have parity zero,
• (P2) Second move: two cancelling crossings in an R₂ move sum to zero,
• (P3) Third move: three involved crossings in R₃ sum to zero.

A parity functor generalizes this, associating to each diagram its own coefficient group, and to each move a group isomorphism acting on the crossing labels such that the axioms remain satisfied in this variable group context (Nikonov, 2021, Ilyutko et al., 2011).

In free and virtual knot theory, the Gaussian parity distinguishes crossings linked with an even vs.\ odd number of chords. For knots in a surface, parities can be homotopical (group elements of $\pi_1(S,z)/\langle[K]\rangle$) or homological (classes in $H_1(S;\mathbb Z)/\langle[K]\rangle$).

In Lorentzian geometry, parity is realized by the deck transformation on sheets of a double cover arising from ring singularities, with each crossing of a ring implementing a nontrivial involution (Rahman, 6 Dec 2025).

For the amplituhedron, parity is encoded in the constancy mod 2 of the crossing number $c_{n,k,m}(Y,\mathcal Z)$, counting topologically nontrivial cells meeting the origin in the quotient vector space $V_Y$ (Blot et al., 2022).

2. The Parity-of-Crossings Theorem: Statement and Variants

The core theorem states that the sum—weighted or unweighted—of the parities assigned to the crossings along the boundary of any polygonal region in the diagram vanishes (additively) or is neutral (multiplicatively) in the coefficient group. In formalism:

• For any oriented parity functor $(P, \mathcal G)$ on a knot diagram $D$ that yields a cell decomposition, and any $k$-gon cell $\pi$ with ordered boundary crossings $v_1,\ldots,v_k$ and incidence signs $\epsilon_\pi(v_i) \in \{\pm1\}$,

$$\sum_{i=1}^k \epsilon_\pi(v_i) \cdot P_D(v_i) = 0 \text{ in } \mathcal G(D),$$

or, for multiplicative notation,

$\prod_{i=1}^k P_D(v_i)^{\epsilon_\pi(v_i)} = 1.$

This polygon relation is deduced inductively from the Reidemeister axioms: the cases $k=1,2,3$ correspond directly to (P1), (P2), and (P3), and the $k>3$ case reduces to those by local insertions and relations (Nikonov, 2021, Ilyutko et al., 2011).

In amplituhedron theory, the theorem manifests as the crossing-constancy theorem: within the image of the positive Grassmannian under the amplituhedron map, the crossing number $c_{n,k,m}(Y,\mathcal Z)$ has constant parity modulo 2, independent of $Y$ or $\mathcal Z$. Membership in the amplituhedron is thereby detected by this crossing parity (Blot et al., 2022).

For excised Kerr-type spacetimes, the theorem asserts: an admissible geodesic which crosses a ring singularity $k$ times lifts to the double cover so that the endpoint is related to the start by the deck involution raised to the $k$th power, so only parity mod 2 matters (Rahman, 6 Dec 2025).

3. Universal Oriented Parity Functors and Classification

Two central universal constructions arise:

• Free knots: The oriented Gaussian parity functor $(P^{og},\mathcal G^{og})$ takes values in $\mathbb Z_4$ or $\{0\}$, depending on whether odd chords are present. Parity values are entirely determined by Gaussian evenness/oddness and the number of odd linkage.
• Knots in surfaces: The universal homotopical parity functor $(HP,\tilde\Pi)$, with $\tilde\Pi$ a quotient of the surface's fundamental group, assigns as parity the class of the “left-half” loop at each crossing. Its abelianization corresponds to the classical homological parity.

Universality here means that any other parity functor factors uniquely through these universal constructions (Nikonov, 2021, Ilyutko et al., 2011). In particular, for free knots, the Gaussian parity is essentially the only nontrivial parity; for knots on surfaces, all parities are pullbacks from homological data.

4. Proof Strategies and Examples

Proofs are founded on inductive arguments using diagrammatic moves. For the $k$-polygon relation, base cases are the parity axioms on Reidemeister 1-, 2-, and 3-moves. For larger $k$, decomposition into smaller polygons (via local R2 or R3 insertions) recycles known relations; the intermediate terms cancel after applying the axioms. This structure is algorithmic, showing that all higher-polygon relations derive from elementary local rules (Nikonov, 2021, Ilyutko et al., 2011).

For the amplituhedron, the constancy of the parity of crossing number is deduced by a combination of Cauchy–Binet/Plücker relations, sign-flip arguments for determinant minors, and triangulation antipodality conditions for cell complexes, culminating in the global constancy of modulo-2 crossing count (Blot et al., 2022).

Geometric/topological contexts (e.g., Kerr spacetimes) recast the parity relation in terms of covering space theory: each ring crossing toggles the sheet, so only the total parity matters for the endpoint's location (Rahman, 6 Dec 2025).

Illustrative Example

In a free-knot chord diagram, considering a 4-gon region with crossings assigned parities $2, 1, 3, 2$ in $\mathbb Z_4$ and all incidences $+1$, the sum $2+1+3+2 = 8 \equiv 0 \pmod{4}$, explicitly confirming the theorem's prediction (Nikonov, 2021).

5. Relation to Knot and Link Invariants

The Parity-of-Crossings Theorem underpins several enhancements and generalizations of classical knot invariants:

• Universal Factorization: All other parity functions (and the invariants they define) must factor uniquely through the universal parity constructions; this includes the oriented Gaussian parity for free knots and the homological parity for knots in surfaces (Ilyutko et al., 2011, Nikonov, 2021).
• Odd Linking Invariant: For free knots, the two odd classes combine into a $\mathbb Z$-valued invariant $L_{odd}(\mathcal K)=||O'| − |O''||$ (Nikonov, 2021).
• Parity-graded Homology: The parity framework enables refinements of Khovanov homology, splitting the complex into even/odd summands and yielding strictly stronger invariants than the classical version (Krylov et al., 2010).
• Minimality and State Sums: Minimal diagram theorems for virtual, free, or flat knots depend on the parity structure—diagrams minimizing the number of nonzero parity crossings minimize the diagram itself (Ilyutko et al., 2011). Parity-refined bracket polynomials include only even crossings in smoothing, giving rise to more discriminating invariants.
• Obstruction Results: On the 2-sphere $S^2$, the theorem confirms any parity functor is trivial; hence all parities of classical knots are trivial, matching the absence of nontrivial parity-based refinements for classical knot theory (Ilyutko et al., 2011, Nikonov, 2021).

The table below summarizes key applications across different theories:

Context	Universal Parity	Key Invariant Constructed
Free knots	Oriented Gaussian ($\mathbb Z_4$)	Odd-linking invariant, minimality
Knots in orientable surface	Homotopical/homological	Index polynomials, parity bracket
Virtual knots	Gaussian parity ($\mathbb Z_2$)	Parity-refined bracket, link invariants
Amplituhedron geometry	Crossing number parity	Characterization of amplituhedron membership
Kerr-type spacetime topology	Deck involution parity	Sheet-determining sector via crossing count

6. Extensions, Analogues, and Further Developments

Parity-of-crossings structures extend beyond knot theory:

• In amplituhedra, the crossing-parity detects whether a given Grassmannian image point lies in the amplituhedron, with explicit combinatorial formulas depending on $k$ and $m$ (Blot et al., 2022).
• In ringed Lorentzian manifolds, the parity of ring crossings fully governs the covering structure of geodesics, with the sheet determined by crossing number modulo 2. This generalizes to multiple rings, infinite chains, and drives the sector structure for globally consistent geodesic histories (Rahman, 6 Dec 2025).
• The formalism of parity functors sets a categorical and functorial framework for constructing invariants and organizing local-to-global diagram relations (Nikonov, 2021, Ilyutko et al., 2011).

These generalizations illustrate the utility and conceptual power of the parity-of-crossings principle across algebraic, combinatorial, and geometric settings, providing a universal algebraic language for local-to-global invariance and classification.

7. Significance and Modern Impact

The Parity-of-Crossings Theorem acts as a unifying algebraic constraint across a spectrum of contemporary mathematical theories, ensuring local rules for crossings and moves imply global invariants and selection rules. Its universality drives the development of both new invariants (e.g., parity-bracket polynomials, parity Khovanov homology) and structural minimality/obstruction results, and underlies topological features in both discrete diagrammatic and continuous geometric models. Its pervasive influence shapes the modern formalism of knot theory, quantum topology, and geometric models of scattering amplitudes, as well as global Lorentzian geometry (Nikonov, 2021, Krylov et al., 2010, Ilyutko et al., 2011, Blot et al., 2022, Rahman, 6 Dec 2025).