On the Links–Gould Invariant of Links

Abstract

We introduce and study in detail an invariant of (1, 1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U q [gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration.

Algebraic links and skein relations of the Links-Gould invariant

“…coincides with the Links-Gould invariant in [4], where α is the complex parameter which represents a family of representations of U q [gl(2|1)]. For the details we refer the reader to [4].…”

“…Note that the connected sum of oriented links depends upon which components are used to produce the connected sum. The property (3) is valid whatever components are selected, which easily follows through the evaluation of the (1,1)-tangle form [4]. For the properties (2),(4),(5), we refer the reader to [4,3].…”

“…Kauffman and J.R. Links [4] showed that the Links-Gould invariant is powerful through evaluation of some links. In particular, D. De Wit [2] showed that the invariant is complete for all prime knots of up to 10 crossings.…”

A skein relation for the HOMFLYPT polynomials of two-cable links

“…They are (1) On the other hand, in 1992, J R Links and M D Gould [21] discovered a 2-variable polynomial invariant for an oriented link, which we call the LG polynomial. It is known that mutant links share the same LG polynomial (De Wit-Links-Kauffman [4]) and all prime knots with less than or equal to 10 crossings are completely classified by the LG polynomial (De Wit [2]). By using skein relations given by De Wit et al [4] and Ishii [7], Ishii and the author [8] gave several examples of different knots and links sharing the same LG polynomials; the smallest such example is a pair of 2-bridge knots with 14 crossings; see Example 5.4.…”

Twisting invariance of link polynomials derived from ribbon quasi-Hopf algebras

“…In particular, the type I quantum superalgebras consisting of U q (gl(m͉n)) and U q (osp(2͉2n)) admit one-parameter families of typical representations which give rise to two-variable link invariants in a natural way. [12][13][14] The work of Reshetikhin and Turaev 15 introduced further the notion of a ribbon Hopf algebra as a particular type of quasi-triangular Hopf algebra. All the quantum algebras fall into the class of ribbon Hopf algebras.…”