History
The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923, but other knot polynomials were not found until almost 60 years later.
In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial.
Soon after Jones' discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state-sum model, which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics.
In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory. Viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable "change of variables").
In recent years, the Alexander polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Ozsváth and Szabó is the Alexander polynomial.
Read more about this topic: Knot Polynomial
Famous quotes containing the word history:
“The myth of independence from the mother is abandoned in mid- life as women learn new routes around the mother—both the mother without and the mother within. A mid-life daughter may reengage with a mother or put new controls on care and set limits to love. But whatever she does, her child’s history is never finished.”
—Terri Apter (20th century)
“Perhaps universal history is the history of the diverse intonation of some metaphors.”
—Jorge Luis Borges (1899–1986)
“The history of all hitherto existing society is the history of class struggles.”
—Karl Marx (1818–1883)