knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating

prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non- trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be ...

s have 8 crossings (and there are three such: 8₁₉, 8₂₀, 8₂₁). It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in knot theory and

3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...

theory, due to their

complement Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visu ...

s having useful and interesting geometric and topological properties. This led

Ralph Fox Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played ...

to ask, "What is an alternating knot?" By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots. In November 2015, Joshua Evan Greene published a preprint that established a characterization of alternating links in terms of definite spanning surfaces, i.e. a definition of alternating links (of which alternating knots are a special case) without using the concept of a

link diagram In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest kn ...

. Various geometric and topological information is revealed in an alternating diagram. Primeness and splittability of a link is easily seen from the diagram. The crossing number of a reduced, alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait conjectures. An alternating

knot diagram In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest k ...

is in one-to-one correspondence with a

planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...

. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner.

Tait conjectures

The Tait conjectures are: #Any reduced diagram of an alternating link has the fewest possible crossings. #Any two reduced diagrams of the same alternating knot have the same writhe. #Given any two reduced alternating diagrams D₁ and D₂ of an oriented, prime alternating link: D₁ may be transformed to D₂ by means of a sequence of certain simple moves called '' flypes''. Also known as the Tait flyping conjecture. Accessed: May 5, 2013.

Morwen Thistlethwaite Morwen Bernard Thistlethwaite (born 5 June 1945) is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory. Biography Mo ...

Louis Kauffman Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, mathematical physicist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. He doe ...

and K. Murasugi proved the first two Tait conjectures in 1987 and

and William Menasco proved the Tait flyping conjecture in 1991.

Hyperbolic volume

Menasco The Menasco Manufacturing Company was an American aircraft engine and component manufacturer. History The company was organized by Albert S. Menasco in 1926 to convert World War I surplus Salmson Z-9 water-cooled nine-cylinder radials into ai ...

, applying Thurston's

hyperbolization theorem In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of the hyperbolization theorem state ...

for

Haken manifold In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in whi ...

s, showed that any prime, non-split alternating link is

hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

, i.e. the link complement has a

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, unless the link is a torus link. Thus hyperbolic volume is an invariant of many alternating links. Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of ''twist regions'' of a reduced, alternating diagram.

References

External links

* *
Celtic Knotwork
to build an alternating knot from its planar graph {{Knot theory, state=collapsed Knot invariants

Tait conjectures

Hyperbolic volume

References

Further reading

External links