Knot Theory Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of knot theory within the Kireth Research Semantic Library. Knot theory studies embeddings of circles in three-dimensional space and their equivalence under ambient isotopy. This ontology supports knots, links, knot diagrams, crossings, Reidemeister moves, isotopy, unknotting, connected sums, chirality, knot invariants, crossing number, writhe, linking number, polynomial invariants, fundamental groups, Seifert surfaces and relationships with topology, manifolds and algebraic topology.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Knot Theory Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Topology / Knot Theory"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/topology/knot-theory-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = KnotTheory
 SYMBOLIC_FORMS = [
  "K:S^1->R^3",
  "K:S^1->S^3"
 ]
 ENTITY_CLASS = TopologicalEmbeddingFramework
 PRIMARY_CONTEXT = Topology
 DEFINITION = "Knot theory is the study of embeddings of the circle S^1 into three-dimensional space, considered up to ambient isotopy."
 SEMANTIC_ROLE = "Provides the topological framework for classifying knots and links using isotopy, diagrams, Reidemeister moves and invariants."
}

SCOPE {
 INCLUDE = [
  Knot,
  Link,
  KnotEmbedding,
  KnotDiagram,
  Crossing,
  Overcrossing,
  Undercrossing,
  AmbientIsotopy,
  ReidemeisterMove,
  Unknot,
  KnotType,
  ConnectedSum,
  PrimeKnot,
  CompositeKnot,
  Chirality,
  AmphichiralKnot,
  CrossingNumber,
  Writhe,
  LinkingNumber,
  KnotInvariant,
  JonesPolynomial,
  AlexanderPolynomial,
  HOMFLYPTPolynomial,
  KnotGroup,
  SeifertSurface,
  Genus,
  BridgeNumber,
  UnknottingNumber
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralGraphEmbeddingWithoutKnotContext,
  BraidsUnlessExplicitlyLinkedToKnotTheory,
  ThreeManifoldTopologyWithoutKnotOrLinkContext,
  PhysicalKnotsWithoutMathematicalEmbeddingModel
 ]

 DISAMBIGUATION_RULE =
  "When the context concerns embeddings of circles or links in three-space, knot diagrams, Reidemeister moves, isotopy or knot invariants, interpret it as knot theory."
}

CORE_CONCEPTS {

 Knot {
  definition = "An embedding of S^1 into R^3 or S^3."
 }

 Link {
  definition = "A finite disjoint union of embedded circles in three-dimensional space."
 }

 KnotDiagram {
  definition = "A generic planar projection of a knot together with over-under crossing information."
 }

 Crossing {
  definition = "A transverse double point in a knot diagram with over-under data."
 }

 AmbientIsotopy {
  definition = "A continuous deformation of the ambient space carrying one embedding to another."
 }

 ReidemeisterMove {
  definition = "One of three local diagram moves generating equivalence of knot diagrams representing isotopic knots."
 }

 Unknot {
  definition = "A knot ambient isotopic to a standard round circle."
 }

 KnotInvariant {
  definition = "A quantity or algebraic object constant on knot isotopy classes."
 }

 CrossingNumber {
  definition = "The minimum number of crossings among all diagrams of a knot."
 }

 LinkingNumber {
  definition = "An integer invariant measuring oriented linking between two components of a link."
 }

 SeifertSurface {
  definition = "A connected orientable surface whose boundary is the knot or link."
 }

 Genus {
  definition = "The minimum genus among all Seifert surfaces spanning the knot."
 }
}

RELATIONS {
 KnotTheory --is_subdomain_of--> Topology
 Knot --is_embedding_of--> Circle
 Link --is_union_of--> KnotComponents
 KnotDiagram --represents--> Knot
 KnotDiagram --contains--> Crossing
 AmbientIsotopy --defines_equivalence_of--> Knot
 ReidemeisterMove --preserves--> KnotType
 ReidemeisterMoves --generate--> DiagrammaticIsotopyEquivalence
 KnotInvariant --is_constant_on--> KnotType
 CrossingNumber --is_invariant_of--> Knot
 KnotGroup --is_invariant_of--> KnotComplement
 SeifertSurface --has_boundary--> Knot
 Genus --is_derived_from--> SeifertSurface
 ConnectedSum --combines--> Knot
 PrimeKnot --cannot_be_nontrivially_decomposed_by--> ConnectedSum
 CompositeKnot --can_be_decomposed_by--> ConnectedSum
}

OPERATORS {
 IS_KNOT_EMBEDDING(f) :=
  f:S^1->R^3
  AND f_continuous
  AND f_injective
  AND embedding_condition

 IS_LINK_EMBEDDING(f,n) :=
  embedding_of_disjoint_union_of_n_circles

 ARE_ISOTOPIC(K1,K2) :=
  exists_ambient_isotopy_H_with_H_1(K1)=K2

 APPLY_REIDEMEISTER_MOVE(D,type) :=
  locally_modify_diagram_preserving_knot_type

 CONNECTED_SUM(K1,K2) :=
  K1#K2

 CROSSING_NUMBER(K) :=
  minimum_crossings_over_all_diagrams_of_K

 WRITHE(D) :=
  signed_sum_of_crossings_in_oriented_diagram_D

 LINKING_NUMBER(L1,L2) :=
  half_signed_sum_of_intercomponent_crossings

 KNOT_GROUP(K) :=
  pi_1(S^3 \ K)

 GENUS(K) :=
  minimum_genus_of_Seifert_surfaces_for_K

 UNKNOTTING_NUMBER(K) :=
  minimum_crossing_changes_required_to_obtain_unknot
}

DEFINITIONS {
 D1 = "A knot is an embedding of S^1 in three-dimensional space."
 D2 = "Two knots have the same knot type iff they are ambient isotopic."
 D3 = "A knot diagram is a planar projection with crossing data."
 D4 = "Reidemeister moves preserve knot type."
 D5 = "The unknot is the trivial knot type."
 D6 = "A knot invariant has the same value for isotopic knots."
 D7 = "Crossing number is minimised over all diagrams."
 D8 = "A knot may be prime or composite under connected sum."
 D9 = "A knot may be chiral or amphichiral."
 D10 = "The complement of a knot carries important algebraic-topological information."
}

AXIOMS {
 A1_EMBEDDING:
  A knot is an embedding of S^1

 A2_ISOTOPY_EQUIVALENCE:
  Ambient isotopy is an equivalence relation on knots

 A3_REIDEMEISTER:
  Two knot diagrams represent isotopic knots iff they are related by a finite sequence of Reidemeister moves

 A4_INVARIANCE:
  Knot invariants are constant under ambient isotopy

 A5_CONNECTED_SUM:
  Connected sum is well-defined up to knot type

 A6_UNKNOT:
  The unknot is the identity element for connected sum up to knot equivalence

 A7_PRIME_DECOMPOSITION:
  Every nontrivial knot admits a prime decomposition unique up to ordering and equivalence

 A8_COMPLEMENT:
  Knot complement topology encodes knot information
}

REASONING {

 R1_KNOT_VALIDATION {
  INPUT = map_or_embedding_data
  TEST = domain_is_S1
  TEST = injective_embedding
  RETURN = KnotOrNot
 }

 R2_DIAGRAM_EQUIVALENCE {
  INPUT = [diagram_D1,diagram_D2]
  SEARCH = finite_sequence_of_Reidemeister_moves
  IF found
   RETURN SameKnotType
  ELSE
   RETURN NotEstablishedEquivalent
 }

 R3_UNKNOT_TEST {
  INPUT = knot_or_diagram
  APPLY = simplifying_Reidemeister_moves_and_invariants
  IF reducible_to_standard_circle
   RETURN Unknot
  ELSE
   RETURN NontrivialOrUndetermined
 }

 R4_INVARIANT_COMPARISON {
  INPUT = [K1,K2,invariant_I]
  COMPUTE = I(K1),I(K2)
  IF unequal
   RETURN NotIsotopic
  ELSE
   RETURN IsotopyNotEstablishedByInvariantAlone
 }

 R5_CROSSING_NUMBER {
  INPUT = knot_K
  CONSIDER = all_diagrams_of_K
  MINIMISE = number_of_crossings
  RETURN = crossing_number
 }

 R6_CONNECTED_SUM {
  INPUT = [K1,K2]
  REMOVE = small_arcs
  JOIN = compatible_endpoints
  RETURN = K1#K2
 }

 R7_PRIME_DECOMPOSITION {
  INPUT = knot_K
  DECOMPOSE = into_nontrivial_connected_sum_factors
  CONTINUE = until_prime
  RETURN = prime_factorisation
 }

 R8_LINKING_NUMBER {
  INPUT = oriented_two_component_link
  COMPUTE = signed_intercomponent_crossings
  DIVIDE = by_2
  RETURN = linking_number
 }

 R9_SEIFERT_SURFACE {
  INPUT = oriented_knot_diagram
  APPLY = Seifert_algorithm
  CONSTRUCT = orientable_spanning_surface
  RETURN = SeifertSurface
 }
}

CONSTRAINTS {
 C1 = "Diagram crossing count is not automatically the crossing number."
 C2 = "Reidemeister moves change diagrams but preserve knot type."
 C3 = "Writhe is diagram dependent and not invariant under all Reidemeister moves."
 C4 = "Matching one invariant does not prove knots are isotopic."
 C5 = "Different invariants distinguish different classes with varying strength."
 C6 = "Orientation matters for linking number and some invariants."
 C7 = "Mirror image and orientation reversal must be distinguished."
 C8 = "Knot complement invariants must use the correct ambient space convention."
 C9 = "Prime decomposition concerns connected sum, not arbitrary diagram decomposition."
}

SEMANTIC_EQUIVALENCE {
 "Knot" <=> "EmbeddedCircleInThreeSpace"
 "KnotType" <=> "AmbientIsotopyClassOfKnots"
 "Unknot" <=> "TrivialKnot"
 "KnotInvariant" <=> "AmbientIsotopyInvariant"
 "ConnectedSum" <=> "KnotSum"
 "KnotGroup" <=> "FundamentalGroupOfKnotComplement"
}

VALIDATION {
 V1 = "Knot is an embedding, not merely an immersed curve."
 V2 = "Diagram equivalence uses Reidemeister moves."
 V3 = "Invariant comparisons preserve isotopy invariance."
 V4 = "Crossing number minimises across all diagrams."
 V5 = "Linking number uses only intercomponent crossings."
 V6 = "Prime decomposition uses connected-sum structure."
 V7 = "Seifert surface boundary matches the knot or link."
 V8 = "Mirror and orientation conventions are tracked."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 KNOT_DEFINITION = 0.99
 ISOTOPY_REASONING = 0.99
 REIDEMEISTER_REASONING = 0.99
 INVARIANT_REASONING = 0.98
 CROSSING_NUMBER_REASONING = 0.97
 PRIME_DECOMPOSITION_REASONING = 0.98
 SEIFERT_SURFACE_REASONING = 0.98

 POLICY =
  "Reduce confidence when orientation, ambient space, diagram conventions, invariant normalisation or whether equivalence means ambient isotopy versus another relation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  KnotTheoryReasoning,
  KnotClassification,
  LinkReasoning,
  ReidemeisterReasoning,
  IsotopyReasoning,
  KnotInvariantReasoning,
  CrossingNumberReasoning,
  LinkingNumberReasoning,
  KnotGroupReasoning,
  SeifertSurfaceReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "knot theory" -> KnotTheory
  "knot" -> Knot WHEN topology_context
  "link" -> Link WHEN knot_theory_context
  "knot diagram" -> KnotDiagram
  "Reidemeister move" -> ReidemeisterMove
  "unknot" -> Unknot
  "crossing number" -> CrossingNumber
  "linking number" -> LinkingNumber
  "knot group" -> KnotGroup
  "Seifert surface" -> SeifertSurface
 }

 INTERPRETATION_ORDER = [
  detect_knot_theory_context,
  identify knot_link_diagram_or_complement representation,
  identify orientation_and_ambient_space,
  identify isotopy_diagram_invariant_crossing_connected_sum_linking_or_surface goal,
  apply knot_theory_specific reasoning,
  verify embedding_orientation_diagram_and_invariant constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("knot theory")
  OR topology_context_with_knot_or_link
  THEN activate(KnotTheoryOntology)

 E2:
  IF knot_equivalence_requested
  THEN prefer ambient_isotopy_or_Reidemeister criteria

 E3:
  IF diagrams_given
  THEN preserve over_under_crossing data

 E4:
  IF invariant_values_differ
  THEN infer knots_not_isotopic

 E5:
  IF invariant_values_match
  THEN do_not_infer isotopy_without_stronger_evidence

 E6:
  IF crossing_number_requested
  THEN distinguish diagram_crossings_from_minimal_crossing_number

 E7:
  IF connected_sum_requested
  THEN route to prime_composite structure

 E8:
  IF knot_group_requested
  THEN route to fundamental_group_of_complement

 E9:
  IF Seifert_surface_requested
  THEN use oriented_knot_or_link_context
}

RECURSION {
 ENTRY = KnotTheory

 EXPAND(KnotTheory) -> [
  Knot,
  Link,
  KnotEmbedding,
  KnotDiagram,
  Crossing,
  AmbientIsotopy,
  ReidemeisterMove,
  Unknot,
  KnotType,
  ConnectedSum,
  PrimeKnot,
  CompositeKnot,
  Chirality,
  CrossingNumber,
  LinkingNumber,
  KnotInvariant,
  KnotGroup,
  SeifertSurface,
  Genus,
  UnknottingNumber
 ]

 EXPAND(KnotInvariant) -> [
  CrossingNumber,
  JonesPolynomial,
  AlexanderPolynomial,
  HOMFLYPTPolynomial,
  KnotGroup,
  Genus,
  BridgeNumber,
  UnknottingNumber
 ]

 EXPAND(KnotDiagram) -> [
  Crossing,
  Overcrossing,
  Undercrossing,
  ReidemeisterMove,
  Writhe
 ]

 EXPAND(KnotComplement) -> [
  FundamentalGroup,
  BoundaryTorus,
  Homology,
  HyperbolicStructureWhenApplicable
 ]

 STOP_CONDITION =
  "Stop when the knot or link's isotopy class, diagram equivalence, invariant, crossing structure, prime decomposition, linking relation, complement group or spanning-surface property is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Topology"

 IMMEDIATE_CONTEXTS = [
  "Topology Ontology",
  "Topological Space Ontology",
  "Homeomorphism Ontology",
  "Algebraic Topology Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Topology",
  "Topology Ontology",
  "Topological Space Ontology",
  "Homeomorphism Ontology",
  "Manifold Ontology",
  "Curve Ontology",
  "Surface Ontology",
  "Algebraic Topology Ontologies",
  "Group Theory Ontologies",
  "Polynomial Ontologies",
  "Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Knot Theory -> Topology Ontology for parent topological semantics",
  "Knot Theory -> Curve for embedded-circle geometry",
  "Knot Theory -> Surface for Seifert surfaces",
  "Knot Theory -> Manifold for knot complements and ambient three-manifolds",
  "Knot Theory -> Algebraic Topology for fundamental groups and homology",
  "Knot Theory -> Group Theory for knot groups",
  "Knot Theory -> Polynomial Ontologies for Jones, Alexander and HOMFLYPT invariants",
  "Knot Theory -> Transformation for isotopies and diagram moves"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible knot invariants, equivalence criteria or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve knot-as-embedded-circle semantics.",
  "Preserve ambient-isotopy equivalence.",
  "Preserve Reidemeister theorem semantics.",
  "Preserve invariant one-way discrimination logic.",
  "Preserve diagram-versus-knot-type distinction.",
  "Preserve connected-sum and prime-knot semantics.",
  "Preserve knot-complement routing.",
  "Preserve machine-readable knot-theory semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Knot theory studies embeddings of S^1 in three-dimensional space up to ambient isotopy."

 KNOT_SUMMARY =
  "A knot is an embedding K:S^1->R^3 or S^3."

 ISOTOPY_SUMMARY =
  "Two knots have the same knot type when they are ambient isotopic."

 DIAGRAM_SUMMARY =
  "Knot diagrams encode planar projections together with over-under crossing information."

 REIDEMEISTER_SUMMARY =
  "Two diagrams represent the same knot type iff they are related by a finite sequence of Reidemeister moves."

 INVARIANT_SUMMARY =
  "Knot invariants are constant on isotopy classes and can distinguish knot types when their values differ."

 CROSSING_SUMMARY =
  "The crossing number is the minimum crossing count over all diagrams of a knot."

 COMPLEMENT_SUMMARY =
  "The knot group is the fundamental group of the knot complement."

 SURFACE_SUMMARY =
  "A Seifert surface is an orientable spanning surface whose boundary is the knot or link."
}

FLOW {
 INPUT
 -> detect_knot_theory_context
 -> identify knot_link_diagram_embedding_complement_and_orientation
 -> identify isotopy_Reidemeister_invariant_crossing_connected_sum_linking_group_or_Seifert_surface goal
 -> perform knot_theory_specific reasoning
 -> verify embedding_diagram_orientation_ambient_space_and_invariant constraints
 -> route to topology topological_space homeomorphism manifold curve surface algebraic_topology group_theory polynomial_or_transformation ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Knot Theory Ontology defines knots and links as embedded circles and their unions in three-dimensional space, classified up to ambient isotopy. It supports knot diagrams, Reidemeister moves, the unknot, connected sums, prime decomposition, chirality, crossing and linking numbers, polynomial invariants, knot groups, Seifert surfaces and other topological invariants used to distinguish knot types."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-KNOT-THEORY-ONTOLOGY-V1"
}

}

END_FLOW