Statements (131)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
knot group theory concept unit of speed |
| gptkbp:abbreviation |
gptkb:kts
kt |
| gptkbp:alternativeName |
Knot
binding_knot decorative_knot knot_group knot_invariant link_(knot_theory) prime_knot torus_knot коса |
| gptkbp:canBeEmbeddedIn |
gptkb:3-dimensional_Euclidean_space
|
| gptkbp:category |
gptkb:topology
low-dimensional topology binding knots nautical unit speed unit |
| gptkbp:characterizedBy |
two coprime integers p and q
|
| gptkbp:class |
by isotopy class
by linking number by number of components |
| gptkbp:component |
knot
link component |
| gptkbp:containsGenus |
(p-1)(q-1)/2
|
| gptkbp:contrastsWith |
gptkb:composite_knot
|
| gptkbp:definedIn |
gptkb:International_Hydrographic_Organization
gptkb:International_Bureau_of_Weights_and_Measures a collection of knots which do not intersect but may be linked together a nontrivial knot that cannot be written as the knot sum of two nontrivial knots |
| gptkbp:defines |
the fundamental group of the complement of a knot in 3-dimensional space
|
| gptkbp:enumerated_in |
knot tables
|
| gptkbp:equivalentTo |
1 nautical mile per hour
1.15078 miles per hour 1.852 kilometers per hour |
| gptkbp:example |
gptkb:granny_knot
gptkb:reef_knot gptkb:square_knot gptkb:surgeon's_knot gptkb:thief_knot gptkb:Borromean_rings gptkb:constrictor_knot gptkb:miller's_knot gptkb:Hopf_link gptkb:cinquefoil_knot_(T(2,5)) gptkb:trefoil_knot_(T(2,3)) gptkb:Whitehead_link gptkb:trefoil_knot figure-eight knot trefoil knot group is non-abelian unknot group is the infinite cyclic group |
| gptkbp:field |
gptkb:topology
knot theory |
| gptkbp:first_classified_by |
gptkb:Peter_Guthrie_Tait
|
| gptkbp:has_crossing |
link crossing
|
| gptkbp:has_diagram |
link diagram
|
| gptkbp:hasAlexanderPolynomial |
(t^{pq}-1)(t-1)/(t^p-1)(t^q-1)
|
| gptkbp:hasApplication |
biology
chemistry mathematical research physics biology (DNA knotting) chemistry (molecular knots) |
| gptkbp:hasCrossingNumber |
(p-1)q if p<q
|
| gptkbp:hasInvariant |
gptkb:Jones_polynomial
gptkb:HOMFLY_polynomial gptkb:Alexander_polynomial knot linking number |
| gptkbp:hasJonesPolynomial |
known for specific p and q
|
| gptkbp:hasMirrorImage |
gptkb:torus_knot_T(q,p)
|
| gptkbp:hasProperty |
nontrivial if both p and q are greater than 1
can be oriented or unoriented can have multiple components cannot be decomposed into simpler knots irreducible under connected sum unique prime decomposition (up to order) |
| gptkbp:hasSpecialCase |
unknot if p or q equals 1
|
| gptkbp:historicalDefinition |
distance between knots on a rope thrown from a ship
|
| gptkbp:introduced |
gptkb:Kurt_Reidemeister
|
| gptkbp:isPrimeKnot |
true if p,q>1
|
| gptkbp:locatedOn |
surface of a torus
|
| gptkbp:measures |
meter per second
|
| gptkbp:minimum_crossing_number |
3 (for trefoil knot)
4 (for figure-eight knot) |
| gptkbp:not_SI_unit |
true
|
| gptkbp:notation |
T(p,q)
π₁(S³ \\ K) |
| gptkbp:origin |
log-line method
measurement of speed at sea |
| gptkbp:property |
can be used to tie bandages
can be used to tie bundles can be used to tie packages secures objects by wrapping and tying can distinguish some but not all knots invariant under ambient isotopy |
| gptkbp:relatedConcept |
gptkb:Brunnian_link
gptkb:knot_(mathematics) unlink split link |
| gptkbp:relatedTo |
gptkb:fundamental_group
gptkb:Wirtinger_presentation gptkb:knot_complement prime number (by analogy) |
| gptkbp:studiedBy |
gptkb:mathematician
3-dimensional space topologists |
| gptkbp:studiedIn |
gptkb:topology
knot theory |
| gptkbp:subclassOf |
knot theory
|
| gptkbp:symbol |
kn
|
| gptkbp:symmetry |
cyclic symmetry
|
| gptkbp:used_in |
gptkb:crafts
climbing sailing scout |
| gptkbp:usedFor |
binding objects together
|
| gptkbp:usedIn |
aviation
meteorology maritime navigation distinguishing knots |
| gptkbp:visualizes |
embedding of circles in 3-space
|
| gptkbp:bfsParent |
gptkb:Brezel
gptkb:butter gptkb:Catholic_Church gptkb:Soft_pretzel gptkb:Pretzel |
| gptkbp:bfsLayer |
4
|