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Concept
Chiral knot
Summary
In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible.
There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.
The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral, and this was proven by Max Dehn in 1914. P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphicheiral knots in the late 1910s. But a counterexample to Tait's conjecture, a 15-crossing amphicheiral knot, was found by Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks in 1998. However, Tait's conjecture was proven true for prime, alternating knots.
Image:TrefoilKnot-02.png|The left-handed trefoil knot.
Image:TrefoilKnot_01.svg|The right-handed trefoil knot.
The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All nontrivial torus knots are chiral. The Alexander polynomial cannot distinguish a knot from its mirror image, but the Jones polynomial can in some cases; if Vk(q) ≠ Vk(q−1), then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant that can fully detect chirality.
A chiral knot that can be smoothly deformed to itself with the opposite orientation is classified as a invertible knot. Examples include the trefoil knot.
If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot, for example the 9 32 knot.
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Crossing number (knot theory)
In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot. It is a knot invariant. By way of example, the unknot has crossing number zero, 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.
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984.
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients. Suppose we have an oriented link , given as a knot diagram. We will define the Jones polynomial, , using Louis Kauffman's bracket polynomial, which we denote by . Here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients.
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