In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l.
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e. diagrams representing the same knot have the same polynomial. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a quantum invariant.
The name HOMFLY combines the initials of its co-discoverers: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter.[1] The addition of PT recognizes independent work carried out by Józef H. Przytycki and Paweł Traczyk.[2]
^Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W.B.R.; Millett, K.; Ocneanu, A. (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S0273-0979-1985-15361-3.
^Józef H. Przytycki; .Paweł Traczyk (1987). "Invariants of Links of Conway Type". Kobe J. Math. 4: 115–139. arXiv:1610.06679.
Alexander polynomial and the Jones polynomial, both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLYpolynomial is also a...
Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLYpolynomial. Soon after Jones' discovery...
unknot. HOMFLYpolynomial Alexander polynomial Volume conjecture Chern–Simons theory Quantum group Jones, Vaughan F.R. (1985). "A polynomial invariant...
{\displaystyle L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.} The HOMFLYpolynomial of the trefoil is L ( α , z ) = − α 4 + α 2 z 2 + 2 α 2 . {\displaystyle...
Relationship". MathWorld. Morton, Hugh R.; Lukac, Sascha G. (2003), "HOMFLYpolynomial of decorated Hopf link", Journal of Knot Theory and Its Ramifications...
algorithms for estimating quantum topological invariants such as Jones and HOMFLYpolynomials, and the Turaev-Viro invariant of three-dimensional manifolds. In...
to Chern–Simons gauge theories for SO(N) in the same way that the HOMFLYpolynomial is related to Chern–Simons gauge theories for SU(N). Kauffman, Louis...
mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not...
There are polynomial-time algorithms for the computation of the Alexander polynomial of a knot. The Jones polynomial, the HOMFLYpolynomial and Reshetikhin–Turaev...
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander...
the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial, HOMFLYpolynomial, and Kauffman polynomial. Jones was...
theory to DNA structure; his initial is the "M" in the name of the HOMFLYpolynomial. Millett graduated from the Massachusetts Institute of Technology...
Its Alexander polynomial is Δ ( t ) = t 2 − t + 1 − t − 1 + t − 2 {\displaystyle \Delta (t)=t^{2}-t+1-t^{-1}+t^{-2}} , its Conway polynomial is ∇ ( z ) =...
Jones polynomial. Also known as the Kauffman bracket. Conway polynomial uses Skein relations. Homflypolynomial or HOMFLYPT polynomial. Jones polynomial assigns...
efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot. It can be difficult to...
the converse is not true. The HOMFLYpolynomial is even better at detecting chirality, but there is no known polynomial knot invariant that can fully...
Alexander polynomials. A variant of the Alexander polynomial, the Alexander–Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish...
particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants...
Meunier-Guttin-Cluzel, S.; Letellier, C. (1999). "Computer evaluation of Homflypolynomials by using Gauss codes, with a skein-template algorithm". Applied Mathematics...
Khovanov homology Genus Knot group Link group Linking no. Polynomial Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime list Stick no. Tricolorability...
Morwen Thistlethwaite in 1987, using the Jones polynomial. A geometric proof, not using knot polynomials, was given in 2017 by Joshua Greene. A second...
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