knot theory In the mathematical field of 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 ...

, a

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

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 The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conje ...

, 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 co ...

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 space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

theory, due to their

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

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 the mathematical field of 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 ...

. 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 the mathematical field of 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 ...

is in one-to-one correspondence with a

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

. 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 In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...

. #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 ''

flype In the mathematical theory of knots, a flype 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 comes from a Scots word meaning '' ...

s''. Also known as the Tait flyping conjecture. Accessed: May 5, 2013.

Morwen Thistlethwaite Morwen Bernard Thistlethwaite 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 Morwen Thistlethwaite ...

Louis Kauffman Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, topologist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the ...

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

and

William Menasco William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory. Biography Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the Unive ...

proved the Tait flyping conjecture in 1991.

Hyperbolic volume

Menasco The Menasco Motors 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 air-coole ...

, 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 Thurston's geometrization theor ...

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 w ...

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

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

, i.e. the link complement has a

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, unless the link is a

torus link In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...

. Thus hyperbolic volume is an invariant of many alternating links.

Marc Lackenby Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory. Lackenby studied mathematics at the University of Cambridge beginning in 1990, and earned his ...

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