Crossing number (knot theory) on:
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In the mathematical area of
There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question asks if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot ''K'' should have larger crossing number than ''K'', but this has not been proven.
Additivity of crossing number under knot sum has been proven for special cases, for example if the summands are
knot theory
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 ...
, the crossing number of 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 ...
is the smallest number of crossings of any diagram of the knot. It is a knot invariant.
Examples
By way of example, theunknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
has crossing number zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...
, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.
Tabulation
Tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots thentwist knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite fa ...
s are listed first). The listing goes 31 (the trefoil knot), 41 (the figure-eight knot), 51, 52, 61, etc. This order has not changed significantly since P. G. Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish Mathematical physics, mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', ...
published a tabulation of knots in 1877.
Additivity
There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question asks if the crossing number is additive when taking knot sums. It is also expected that a satellite of a knot ''K'' should have larger crossing number than ''K'', but this has not been proven.
Additivity of crossing number under knot sum has been proven for special cases, for example if the summands are alternating knot
In knot theory, a knot 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 ...
s (or more generally, adequate knot), or if the summands are torus knots. Marc Lackenby has also given a proof that there is a constant ''N'' > 1 such that , but his method, which utilizes normal surface In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a ''triangle'' or a ''quad'' (see figure). A triangle cuts off a vertex of the tetrahedron wh ...
s, cannot improve ''N'' to 1.
Applications in bioinformatics
There are connections between the crossing number of a knot and the physical behavior of DNA knots. For prime DNA knots, crossing number is a good predictor of the relative velocity of the DNA knot in agarosegel electrophoresis
Gel electrophoresis is a method for separation and analysis of biomacromolecules ( DNA, RNA, proteins, etc.) and their fragments, based on their size and charge. It is used in clinical chemistry to separate proteins by charge or size (IEF ...
. Basically, the higher the crossing number, the faster the relative velocity. For composite 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, this does not appear to be the case, although experimental conditions can drastically change the results..
Related invariants
There are related concepts ofaverage crossing number
In the mathematical subject of knot theory, the average crossing number of a knot is the result of averaging over all directions the number of crossings in a knot diagram of the knot obtained by projection onto the plane orthogonal to the directio ...
and asymptotic crossing number
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
. Both of these quantities bound the standard crossing number. Asymptotic crossing number is conjectured to be equal to crossing number.
Other numerical knot invariants include the bridge number
In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.
Definition
Given a knot or link, draw a diagram of the l ...
, linking number, stick number
In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot. Specifically, given any knot K, the stick number of K, denoted by \o ...
, and unknotting number
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n, then there exists a diagram of the k ...
.
References