Narrow Results

An alternative construction of the invariant of homology three spheres valued in a completion of an integral polynomial ring associated to each quantized complex simple Lie algebra by Habiro and Lê [Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol.20 (2016) 2687–2835, doi:10.2140/gt.2016.20.2687, arXiv:1503:03549v1] is given.

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and $\mathrm{\text{Hom}}$ quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

A (t, s)-rack is a rack structure defined on a module over the ring . We identify necessary and sufficient conditions for two (t, s)-racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t, s)-racks and give some computations and examples. As an application, we use these enhanced invariants to obtain obstructions to knot ordering.

We introduce an algebra ℤ[X, S] associated to a pair X, S of a virtual birack X and X-shadow S. We use modules over ℤ[X, S] to define enhancements of the virtual birack shadow counting invariant, extending the birack shadow module invariants to virtual case. We repeat this construction for the twisted virtual case. As applications, we show that the new invariants can detect orientation reversal and are not determined by the knot group, the Arrow polynomial and the Miyazawa polynomial, and that the twisted version is not determined by the twisted Jones polynomial.

As a generalization of quandles, biquandles have given many invariants of classical/surface/virtual links. In this paper, we show that the fundamental quandle $Q(L)$ of any classical/surface link $L$ detects the fundamental biquandle $\mathrm{\text{BQ}}(L)$; more precisely, there exists a functor ${ℬ}$ from the category of quandles to that of biquandles such that ${ℬ}(Q(L))\cong \mathrm{\text{BQ}}(L)$. Then, we can expect invariants from biquandles to be reduced to those from quandles. In fact, we introduce a right-adjoint functor ${𝒬}$ of ${ℬ}$, which implies that the coloring number of a biquandle $X$ is equal to that of the quandle ${𝒬}(X)$.

We study a topological Abelian gauge theory that generalizes the Abelian Chern–Simons one, and that leads in a natural way to the Milnor's link invariant when the classical action on-shell is calculated.

Topologically massive Yang–Mills theory is studied in the framework of geometric quantization. Since this theory has a mass gap proportional to the topological mass m, Yang–Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern–Simons theory with level number k. In this paper, the near Chern–Simons limit is studied where the distance is large enough to give an almost topological theory, with a small contribution from the Yang–Mills term. It is shown that this almost topological theory consists of two copies of Chern–Simons with level number k/2, very similar to the Chern–Simons splitting of topologically massive AdS gravity. Also, gauge invariance of these half-Chern–Simons theories is discussed. As m approaches to infinity, the split parts add up to give the original Chern–Simons term with level k. Reduction of the phase space is discussed in this limit. Finally, a relation between the observables of topologically massive Yang–Mills theory and Chern–Simons theory is shown. One of the two split Chern–Simons pieces is shown to be associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang–Mills theory observables in the near Chern–Simons limit.

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

We define a new combinatorial complex computing the hat version of link Floer homology over ℤ/2ℤ, which turns out to be significantly smaller than the Manolescu–Ozsváth–Sarkar one.

Let L be a link and its link invariant associated with the vector representation of the quantum (super)algebra U_q(A). Let F_L(r, s) be the Kauffman link invariant for L associated with the Birman–Wenzl–Murakami algebra BWM_f(r, s) for complex parameters r and s and a sufficiently large rank f.

For an arbitrary link L, we show that and for each positive integer n and all sufficiently large f, and that and are identical up to a substitution of variables.

For at least one class of links F_L(-r, -s) = F_L(r, s) implying for these links.

We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e^2πi/5, e^-2πi/5) is DQC1-complete and at points for some integer k is in BQP.

To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids.

To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones–Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of are alternating links.

A rack shadow or rack set is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.

In a prequel to this article, we used abstract Wiener measure to define the Chern–Simons path integral over ℝ³. In this sequel, we compute the Wilson Loop observable for the non-abelian gauge group and compare with current knot literature.

The column group is a subgroup of the symmetric group on the elements of a finite birack generated by the column permutations in the birack matrix. We use subgroups of the column group associated to birack homomorphisms to define an enhancement of the integral birack counting invariant and give examples which show that the enhanced invariant is stronger than the unenhanced invariant.

We introduce an associative algebra ℤ[X, S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of ℤ[X, S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by the Alexander polynomial or the unenhanced birack counting invariants.

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(2^g) basis elements, with each coefficient a polynomial with at most n non-zero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by for some C. From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in n times .

We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.

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 ${\mathrm{\text{FTL}}}_{d,n}(q)$.

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.

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.