Narrow Results

We show that two Alexander biquandles M and M′ are isomorphic if and only if there is an isomorphism of ℤ[s^±1, t^±1]-modules h : (1 - st)M → (1 - st)M′ and a bijection g : O_s(A) → O_s(A′) between the s-orbits of sets of coset representatives of M/(1 - st)M and M′/(1 - st)M′ respectively satisfying certain compatibility conditions.

In this paper, we define a new algebro-geometric invariant of three-manifolds resulting from Dehn surgery along a hyperbolic knot complement in S³. We establish a Casson-type invariant for these three-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some three-manifolds resulting from Dehn surgery along the figure-eight knot.

It is shown that the canonical formulation of the Abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled to external sources with support on curves and points in the spatial plane. We explicitly calculate a nontrivial invariant that could be seen as a "projection" of the Milnor's link invariant , and as such, it measures the entanglement of generalized (or projected) Borromeans Rings in the Euclidean plane.

We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a₁, b₁, a₂, b₂, …) and for the first nonfundamental representation R = [2]. Instead of calculating the mixing matrices directly, as suggested [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654], we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1, 1]. The result applies, in particular, to the figure eight knot 4₁, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in H. Itoyama, A. Mironov, A. Morozov and And. Morozov, arXiv:1203.5978.

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation $R=[2,1]$. The project involves several steps:

• ⁽ⁱ⁾parametrization of big families of knots á la [A. Mironov and A. Morozov, arXiv:1506.00339],
• ⁽ⁱⁱ⁾evaluating Racah/mixing matrices for various numbers of strands in various representations á la [A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys.03, 034 (2012), arXiv:1112.2654],
• ⁽ⁱⁱⁱ⁾tabulating and collecting the results at http://www.knotebook.org.

In this paper, we discuss only the representation $R=[2,1]$ and construct all necessary ingredients that allow one to evaluate knot/links represented by three-strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family. This family contains over 80% of knots with up to 10 intersections, but does not include mutants.

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n=6 and n=7 is described. In both of these cases, each knot space consists of five components,but contains only three (when n=6) or four (when n=7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n≥8 will also be discussed.

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalized knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well-known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n ≥ 2. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n ≥ 2, carries more information about K than the isomorphism type of the fundamental group. The appendix contains some results on representations of the trefoil group in PSL(2, p) that are needed for the proof.

We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).

Using these invariants, we prove that left and right trefoils are not isotopic using this invariant (in a particular case).

We show how to use Bar-Natan's "divide and conquer" approach to computation to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

In the 1920's Artin defined the braid group, B_n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q₁ → Q₀. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.

Bing, Bothe and Shilepsky studied knots that were wild at every point, and Bothe developed an invariant for a certain subclass of these knots. This paper develops invariants which distinguish ambient isotopy equivalence classes of these knots (which include the "Bing sling") and shows that there is an uncountable number of inequivalent of Bing sling type knots. As an aside, it is also shown that it is possible for inequivalent wild knots to have homeomorphic complements.

In this short note, we prove that the sphere and 4-tube relations “almost” imply the torus relation in Bar-Natan’s geometric formalism of Khovanov homology theory. As an application, we point out that the invariance and the functoriality can be proved without using the torus relation.

S. V. Duzhin [Decomposable skew-symmetric functions, Mosc. Math. J.3(3) (2003) 881–888] suggested a construction of weight systems from polynomials possessing certain symmetry properties with respect to permutations of their arguments. In particular, he considered a set of polynomials in five variables, which he called $\mathrm{\text{K}}5$. A natural construction gives polynomials in $\mathrm{\text{K}}5$, and he asked, whether there are polynomials in $\mathrm{\text{K}}5$ that are not produced by this construction. In this paper we give a negative answer to Duzhin’s question.

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

The number ${\mathrm{\text{Col}}}_Q(K)$ of colorings of a knot $K$ by a finite quandle $Q$ has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement ${\mathrm{\text{Col}}}_Q^F(K)$ to this knot invariant involving any computable functor $F$ from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor $F={}^{ab}$ that sends each finitely presented group $H$ to its abelianization $H^{ab}=H/[H.H]$. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with $\le 11$ crossings.

Given a knot $K$, we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\ge 2$ these “generalized knot groups” determine $K$ up to reflection.

The second author has shown that for $n\ge 2$, the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues $S{K}_{a,b}=T_{a,b}\#T_{-a,b}$, $G{K}_{a,b}=T_{a,b}\#T_{a,b}$, constructed as connected sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\ge 2$ and $n$ satisfying a coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(S{K}_{a,b})$ and $G_n(G{K}_{a,b})$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\ge 2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group.

We formulate large $N$ duality of $U(N)$ refined Chern–Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $\mathrm{\Omega }$-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in $S^3$ to refined BPS invariants in the resolved conifold. Assuming that the extra $U(1)$ global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

In this paper, we use artificial neural networks to predict and help compute the values of certain knot invariants. In particular, we show that neural networks are able to predict when a knot is quasipositive with a high degree of accuracy. Given a knot with unknown quasipositivity, we use these predictions to identify braid representatives that are likely to be quasipositive, which we then subject to further testing to verify. Using these techniques, we identify 84 new quasipositive 11 and 12-crossing knots. Furthermore, we show that neural networks are also able to predict and help compute the slice genus and Ozsváth-Szabó $\tau$-invariant of knots.

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to $(\ell ,k\ell -1)$ torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.