HOMFLY polynomial

HOMFLY polynomial
Name	HOMFLY polynomial
Variables	"l, m"
Field	Topology
Introduced	1980s
Contributors	"Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter"

HOMFLY polynomial is an invariant of oriented links in three-dimensional space that assigns to each link a Laurent polynomial in two variables, used to distinguish knots and links in Knot theory, Topology, and Low-dimensional topology. It generalizes earlier invariants from Jones polynomial and Alexander polynomial and has been applied across mathematical and physical contexts including Quantum field theory, Statistical mechanics, and Category theory. The polynomial played a role in interactions among researchers affiliated with institutions such as Princeton University, University of California, Los Angeles, and University of Cambridge.

Definition

The invariant is defined for oriented links in the three-sphere S^3 and values lie in the ring of Laurent polynomials over integers in variables often denoted l and m; rigorous formulations were developed by authors at meetings involving participants from University of Illinois, Rutgers University, and Imperial College London. The definition is characterized by normalization conditions that specify the value on the unknot and by compatibility with a skein relation; these conditions parallel normalization choices made in definitions of the Alexander polynomial associated to J. W. Alexander and the Jones polynomial introduced by Vaughan Jones. The invariant is insensitive to ambient isotopy and is therefore constant under Reidemeister moves recognized by researchers at Princeton University and University of California, Berkeley.

Properties

The polynomial satisfies several algebraic and topological properties studied in seminars at Massachusetts Institute of Technology, Harvard University, and University of Oxford. It is multiplicative under split union operations considered in conferences at California Institute of Technology and behaves predictably under orientation reversal examined by teams at Yale University and University of Chicago. Symmetry relations analogous to those for the Alexander–Conway polynomial and relations to degree bounds linked to Seifert surfaces were investigated by groups at ETH Zurich and University of Tokyo. The polynomial detects chirality in many cases, a phenomenon explored by researchers from University of Warwick and University of Toronto.

Computation and skein relation

Computation uses a skein relation first distilled in collaborative work involving scholars at SUNY Stony Brook, University of California, Santa Cruz, and Brown University. The skein relation connects three link diagrams differing at a crossing and resembles relations used for the Jones polynomial at Bell Labs and for the Conway polynomial in lectures at Imperial College London. Practical algorithms employ state sum models and cabling techniques developed at University of Illinois Urbana-Champaign and computational packages from groups at University of Washington and University of Sydney. Complexity analyses and automated tabulations were reported in projects at MPI for Mathematics, Los Alamos National Laboratory, and Cornell University.

Examples and specializations

Specializations recover classical invariants: a one-variable specialization yields the Jones polynomial as observed in work at Rutgers University and University of California, Los Angeles, while another specialization gives the Alexander polynomial analogous to constructions by J. W. Alexander and follow-up studies at University of Cambridge. Explicit computations for torus knots and links, studied by researchers at Princeton University, University of Manchester, and Tokyo Institute of Technology, produce closed formulas aligning with results from Representation theory groups at Institute for Advanced Study. Tables of values for prime knots up to specific crossing numbers were compiled by teams at University of British Columbia, University of Melbourne, and University of Bonn.

Applications and connections

Applications span interactions with Quantum groups developed by researchers at University of Southern California and Institut des Hautes Études Scientifiques, connections to Chern–Simons theory explored by physicists at Institute for Advanced Study and Perimeter Institute, and relations to categorification programs linked to work at University of Oxford and Columbia University. The polynomial informs studies in Statistical mechanics models pursued at Princeton University and serves as a testing ground for homological invariants inspired by projects at Harvard University and Stanford University. It has influenced computational topology initiatives at European Mathematical Society-affiliated centers and algorithmic knot recognition efforts at Microsoft Research and Google DeepMind collaborations.

History and development

The invariant emerged from parallel and independent developments in the 1980s by mathematicians publishing from institutions such as University of Chicago, Brooklyn College, and University of California, Santa Barbara and from collaborative workshops at Mathematical Sciences Research Institute and International Congress of Mathematicians satellite meetings. Key contributions were announced around the same time in papers and preprints circulated through networks including American Mathematical Society and London Mathematical Society venues. Subsequent refinements, computational projects, and extensions were carried out by researchers affiliated with National Academy of Sciences members and groups at École Normale Supérieure and KnotTheory research collectives.

Category:Knot theory