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The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts
Tait_conjectures
Scottish mathematical physicist (1831–1901)
Episcopal Church, Edinburgh. The Tait conjectures are three conjectures made by Tait in his study of knots. The Tait conjectures involve concepts in knot theory
Peter_Guthrie_Tait
Disproven graph theory
In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices"
Tait's_conjecture
properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few
Alternating_knot
Mathematician specializing in knot theory
Murasugi proved the first two Tait conjectures in 1987 and Thistlethwaite and William Menasco proved the Tait flyping conjecture in 1991. Thistlethwaite also
Morwen_Thistlethwaite
Three linked but pairwise separated rings
that is simultaneously alternating and algebraic. It follows from the Tait conjectures that the crossing number of the Borromean rings (the fewest crossings
Borromean_rings
Unsolved problem in graph theory
Tait's and Tutte's conjectures, stating that every bipartite cubic polyhedron is Hamiltonian, or, equivalently, that every counterexample to Tait's conjecture
Barnette's_conjecture
This is a list of notable mathematical conjectures. The following conjectures remain open. The (incomplete) column "cites" lists the number of results
List_of_conjectures
British-Canadian codebreaker and mathematician (1917–2002)
graphs, and Hamiltonian and non-Hamiltonian graphs. He disproved Tait's conjecture, on the Hamiltonicity of polyhedral graphs, by using the construction
W._T._Tutte
Study of mathematical knots
of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually
Knot_theory
formulated what is now known as the Tait conjectures on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved
History_of_knot_theory
Orientable surface whose boundary is a knot or link
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Seifert_surface
Prime knot with crossing number 10
that the writhe of a reduced diagram of a knot is an invariant (see Tait conjectures), as the two diagrams for the pair have different writhes. In some
Perko_pair
Graph with all vertices of degree 3
Tait conjectured that every cubic polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Tait's conjecture, the
Cubic_graph
On Hamiltonian cycles in planar graphs
of the Tutte graph in 1946. Instead, Barnette's conjecture, still unproven, weakens Tait's conjecture in a different way, to bipartite planar graphs.
Tutte's theorem on Hamiltonian cycles
Tutte's_theorem_on_Hamiltonian_cycles
Unique knot with a crossing number of four
theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Polynomials arising in knot theory
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
HOMFLY_polynomial
Simplest non-trivial closed knot with three crossings
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Trefoil_knot
Motif with two doubly-interlinked loops
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Solomon's_knot
Two interlinked loops with five structural crossings
Whitehead, who spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, he used the link as part of his construction of the now-named
Whitehead_link
graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle. Published
Tutte_graph
Simplest nontrivial knot link
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Hopf_link
Function of a knot that takes the same value for equivalent knots
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Knot_invariant
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Prime_knot
Collection of knots that do not intersect, but may be linked
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Link_(knot_theory)
Group whose operation is a composition of braids
topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of the proposed particles anyons. These
Braid_group
Two-variable polynomial knot invariant
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Kauffman_polynomial
Mathematical invariant of a knot or link
would give the hyperbolic volume of the knot complement. (See Volume conjecture.) In 2000 Mikhail Khovanov constructed a certain chain complex for knots
Jones_polynomial
Family of mathematical knots
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Twist_knot
Prime knot named for John Horton Conway
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Conway_knot
Type of knot in knot theory
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2-bridge_knot
How many times curves wind around each other
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Linking_number
3-regular graph with no 3-edge-coloring
snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880, but their name is much newer, given
Snark_(graph_theory)
Invariant of mathematical knots
theorem of Peter Kronheimer and Tomasz Mrowka, formerly known as the Milnor conjecture (see below). There is a spectral sequence relating Khovanov homology with
Khovanov_homology
Property in knot theory
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Tricolorability
Bipartite non-Hamiltonian polyhedral graph
is much larger than the Herschel graph. A refinement of Tait's conjecture, Barnette's conjecture that every bipartite 3-regular polyhedral graph is Hamiltonian
Herschel_graph
Knot invariant
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Alexander_polynomial
On Hamiltonian cycles in planar graphs
instance it can prove non-Hamiltonicity of some counterexamples to Tait's conjecture that cubic polyhedral graphs are Hamiltonian. Grinberg's theorem is
Grinberg's_theorem
Complement of a knot in three-sphere
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Knot_complement
Knot which lies on the surface of a torus in 3-dimensional space
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Torus_knot
Path in a graph that visits each vertex exactly once
permutohedron Subhamiltonian graph, a subgraph of a planar Hamiltonian graph Tait's conjecture (now known false) that 3-regular polyhedral graphs are Hamiltonian
Hamiltonian_path
Knot that is not equivalent to its mirror image
Tait's conjecture, a 15-crossing amphichiral knot, was found by Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks in 1998. However, Tait's conjecture was
Chiral_knot
One of three types of isotopy-preserving local changes to a knot diagram
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Reidemeister_move
Origin and evolution of the symbols used to write equations and formulas
(1860s) led Peter Guthrie Tait, in 1885, to publish a topological table of knots with up to ten crossings known as the Tait conjectures. Tensor calculus was
History of mathematical notation
History_of_mathematical_notation
Type of mathematical knot
homeomorphic if and only if their fundamental groups are isomorphic. Waldhausen conjectured what is now the Jaco–Shalen–Johannson-decomposition of 3-manifolds, which
Satellite_knot
Normalized hyperbolic volume of the complement of a hyperbolic knot
first studied by William Thurston in connection with his geometrization conjecture. A hyperbolic link is a link in the 3-sphere whose complement (the space
Hyperbolic_volume
Struthers 1823–1899 anatomist Peter Guthrie Tait 1831–1901 mathematical physicist proposer of the Tait conjectures in Knot theory Thomas Telford 1757–1834
List_of_Scottish_scientists
Embedding of the circle in three dimensional Euclidean space
Another convenient representation of knot diagrams was introduced by Peter Tait in 1877. Any knot diagram defines a plane graph whose vertices are the crossings
Knot_(mathematics)
Operation on a knot
is a kind of manipulation of knot and link diagrams used in the Tait flyping conjecture. It consists of twisting a part of a knot, a tangle T, by 180 degrees
Flype
Attempt to classify and tabulate all possible knots
the aether. In an attempt to make a periodic table of the elements, P. G. Tait, C. N. Little and others started to attempt to count all possible knots.
Knot_tabulation
Link formed from a finite number of twisted sections
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Pretzel_link
Mathematical knot with crossing number 5
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Three-twist_knot
Polynomial invariant of framed links
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Bracket_polynomial
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List_of_prime_knots
Exception to a proposed general rule
show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable
Counterexample
Link that consists of finitely many unlinked unknots
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Unlink
Mathematical knot with crossing number 7
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7_2_knot
Mathematical knot with crossing number 7
This knot is used to construct the simplest known counterexample to the conjecture that the unknotting number is additive under connected sum. The 71 knot
71_knot
Knot that bounds an embedded disk in 4-space
are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). The conditions locally-flat or smooth are essential in the definition:
Slice_knot
Type of invariant in Knot theory
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Finite_type_invariant
Integer-valued knot invariant; least number of crossings in a knot diagram
knot), 51, 52, 61, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.[non-primary source needed] There
Crossing_number_(knot_theory)
Loop seen as a trivial knot
to the field of knot theory because they can serve as cases for which conjectures about unknotting algorithms can be tested. Early examples of hard unknot
Unknot
Invariant of a knot diagram
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Writhe
Generalization of knots in 3-dimensional Euclidean space
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Virtual_knot
Type of mathematical link
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Hyperbolic_link
Interlinked multi-loop construction where cutting one loop frees all the others
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Brunnian_link
Theorem in formal logic
It was settled positively: By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966); Independently by Prawitz
Takeuti's_conjecture
Fundamental group of a knot complement
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Knot_group
Mathematical tool for studying knots
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Skein_relation
Milnor conjecture (topology) Milnor map Möbius energy Mutation (knot theory) Physical knot theory Planar algebra Smith conjecture Tait conjectures Temperley–Lieb
List_of_knot_theory_topics
Knot that can't be tied in a string of constant diameter
Every closed curve containing a wild arc is a wild knot. It has been conjectured that every wild knot has infinitely many quadrisecants. As well as their
Wild_knot
Notation used to describe knots based on operations on tangles
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Conway_notation_(knot_theory)
Type of mathematical knot
knots and potential counterexamples to the property 2R and slice-ribbon conjectures", Geometry & Topology, 14 (4): 2305–2347, arXiv:1103.1601, doi:10.2140/gt
Ribbon_knot
Mathematical knot with crossing number 5
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Cinquefoil_knot
Knot invariant named after Cahit Arf
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Arf_invariant_of_a_knot
Smallest number of edges of an equivalent polygonal path for a knot
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Stick_number
Mathematical notation for describing the structure of knots
Morwen Thistlethwaite, who refined a notation originally due to Peter Guthrie Tait. It is not an invariant of the associated knot. To generate the Dowker–Thistlethwaite
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite_notation
Analog of the knot group
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Link_group
Mathematical knot with crossing number 7
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
74_knot
Type of mathematical knot
the figure-eight knot, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes. Kirby, R., (1978). "Problems in
(−2,3,7)_pretzel_knot
Graph made from vertices and edges of a convex polyhedron
into smaller convex polygons may be found using the Tutte embedding. Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which
Polyhedral_graph
Determining whether a knot is the unknot
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Unknotting_problem
Planar bipartite graph with 25 vertices and 31 edges
graphs to produce the Tutte graph, the first known counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle. By integrating
Walther_graph
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Knot_polynomial
Connected sum of two trefoil knots with opposite chirality
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Square_knot_(mathematics)
17th-century conjecture proved by Andrew Wiles in 1994
the case with some other past conjectures, such as with Skewes' number, and it could not be ruled out in this conjecture.) The strategy that ultimately
Fermat's_Last_Theorem
trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible
Invertible_knot
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Crosscap_number
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Bridge_number
Mathematical knot with crossing number 6
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62_knot
Every knot or link can be represented as a closed braid
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Alexander's_theorem
Class of mathematical knot with special properties
Gordon conjectured these were the only knots admitting lens space surgeries. This is now known as the Berge conjecture. The Berge conjecture states that
Berge_knot
Connected sum of two trefoil knots with same chirality
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
Granny_knot_(mathematics)
Invariant of framed knots
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Self-linking_number
British-American scholar
California; and California State University, Long Beach. He won the James Tait Black Award for his biography of Christopher Isherwood. Since How It Is:
Brian_Finney
Mathematical knot
knots and potential counterexamples to the property 2R and slice-ribbon conjectures". Geometry & Topology. 14 (4): 2305–2347. arXiv:1103.1601. doi:10.2140/gt
Fibered_knot
graph Ramsey's theorem Sperner's lemma Strong coloring Subcoloring Tait's conjecture Total coloring Uniquely colorable graph Path (graph theory) Seven
List_of_graph_theory_topics
Constructs with triply-connected vertices
absent above if the graph has no Hamiltonian cycle, which is rare (see Tait's conjecture). In this case a list of edges between pairs of vertices labeled 0
Table_of_simple_cubic_graphs
Fibered Knot List of knots and links Open knot theory Ribbon Slice Sum Surgery theory Tait conjectures Twist Wild Writhe Virtual knots Category Commons
List of mathematical knots and links
List_of_mathematical_knots_and_links
Mathematical knot with crossing number 6
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Stevedore_knot_(mathematics)
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