Narrow Results

This text is a brief description of the Jones polynomial as it appears in the announcement in [9], together with Jones’ comments in the letter he addressed to Joan Birman on 31st May 1984 [8]. As much as possible I have followed the author’s initial notations. I add comments about the history of the events as I remember them, with the help of several testimonies of the “actors”. Among them: Didier Hatt-Arnold, Pierre de la Harpe, Cam Van Quach Hongler. It took me nontrivial efforts with my memories to get the chronology right (I hope so). The help of Eva Bayer was decisive. My warmest thanks to her. One may disagree with some of my statements which express a personal opinion. On purpose I have adopted a candid style.

In this paper we present the solutions of the star-triangle equation for Boltzmann weights of the (N_α,N_β) model. This model consists of two scalar Potts models (of N_α and N_β states, respectively) coupled together by four-spin interaction terms and was introduced by Domany and Riedel to study the phase transitions in adsorbed thin films. If N_α,N_β≠2, the star-triangle equations have nontrivial self-dual and non-self-dual solutions only for N_α=N_β; whereas if N_α≠N_β=2, we have other self-dual and non-self-dual solutions. After applying partial duality transformations several of our solutions can be identified with known solutions for Interaction-Round-a-Face (IRF) models. Our results also mean that our high-genus solutions found earlier are not the only solutions to the star-triangle equations in chiral Potts models with six or more states per site.

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.

The solutions of Yang–Baxter equation (YBE) associated with quantum information have been reviewed with new progress. The additivity rule for two particles is shown to obey Lorentzian type other than the Gallilean. Acting the braiding operations on the topological basis, it turns out that the $N$-dimensional representations obeying YBE are nothing but the Wigner’s $D$-functions. The connection between the extremization of ${\ell }_1$-norm as well as of von Neumann entropy and the reduction of YBE to braid relation is also discussed.

We investigate a family of (reducible) representations of the braid groups corresponding to a specific solution to the Yang–Baxter equation. The images of under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of factoring over Temperley–Lieb algebras and the corresponding link invariants.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

This paper is an unpublished version of a talk that the author gave in Brazil in 1987 at a meeting of the Brazilian Mathematical Society, while visiting Sostenes Lins in Recife, Brazil. The paper is an introduction to state models for link invariants, and it gives a surface-genus interpretation of the extreme terms in the bracket polynomial.

A generalization of the Kauffman tangle algebra is given for Coxeter type D_n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A_{n - 1}. The proof involves a diagrammatic version of the Brauer algebra of type D_n of which the generalized Temperley–Lieb algebra of type D_n is a subalgebra.

We introduce a recoupling theory for virtual braided trees. This recoupling theory can be utilized to incorporate swap gates into anyonic models of quantum computation.

Using algebraic approach, we show which elements of a basis (in the reduced form) of Temperley–Lieb algebra can be obtained as Kauffman states of lattice crossing and generalized crossing. In addition, we give an algorithm for finding them and we obtain further results concerning coefficients of the Catalan states that can be obtained as Kauffman states of $m\times 2$ lattice crossing.

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras ${\mathrm{\text{Y}}}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\mathrm{\text{FTL}}}_{d,n}(q)$ which is the appropriate Temperley–Lieb analogue of ${\mathrm{\text{Y}}}_{d,n}(q)$, as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra ${\mathrm{\text{PTL}}}_n(q)$ and we prove an isomorphism of this algebra with a subalgebra of ${\mathrm{\text{FTL}}}_{d,n}(q)$.

In this paper, we construct a new homology theory for semi-groups satisfying the right self-distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one-term and two-term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley–Lieb algebras.

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this invariant is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of nonisotopic links than the original Jones polynomial, such as the Thistlethwaite link from the unlink with two components.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with $U(1)\times U(1)$-equivariant cohomology of ${ℂ\mathbb{P}}^1$. Motivated by the relationship between the Temperley–Lieb algebra and annular Khovanov homology, we also introduce an equivariant analog of the Temperley–Lieb algebra.

We show that the matrix of chromatic joins and the Gram matrix of a Temperley-Lieb algebra are similar.

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

For Temperley–Lieb algebras of types $B$ and $D$, we construct their Gröbner-Shirshov bases and the corresponding standard monomials

We begin the study of the representation theory of the infinite Temperley–Lieb algebra. We fully classify its finite-dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TL${}_n$ that generalizes to give extensions of TL${}_{\infty }$ representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang–Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley–Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang–Baxter equations; and virtual Temperley–Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.