arXiv research

The paper studies Fox pairings of Poincaré duality groups using group cohomology.

We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawa…

The paper proves a generalized Kauffman-Harary conjecture for prime determinant links.

Functorial approach connects operads to Lie bialgebras.

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. W…

Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.

The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.

Researchers study chirality in a specific type of torus-covering link.

Log-concave coefficient sequences for two-bridge knots proved.

Study on knots using 17 colors, finding specific color assignments.

Dihedral linking invariant uses knot colorings to distinguish knots.

A torus-covering $T^2$-knot is a surface-knot of genus one determined from a pair of commutative braids. For a torus-covering $T^2$-knot $F$, we determine the number of irreducible metabelian $SU(2)$-representations of the knot group of $F$ in terms of the knot determinant of $F$. It is similar to the result due to Lin…

We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…

Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.

By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twisting…

New Vassiliev invariant of order three derived from Fox-Hatcher cycles.

Study on Fox's trapezoidal conjecture for specific alternating links.

A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow Σ\times [0,1]$, where $Σ$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classic…

Introduces new spaces for configurations of points with specific monodromies.

Study on coloring virtual tangles with integer and modular arithmetic.

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possi…

We solve a century-old conjecture about Alexander polynomials of special alternating links.

Formula calculates volume of two-bridge knots.

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Let $β=β_2(π;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(π,w)$) there are at most $2^β$ orbits of $k$-invariants determining "strongly minimal" complexes (i.…

Mandelbrot unified diverse fields with scaling concept.

Paper uses Long-Moody construction for new braid group representations.

Characterizes a specific type of alternating knot.

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-com…

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has…

This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria…

This expository paper describes how the knot invariant Fox coloring can be applied to tangles.

Study Alexander polynomials of special alternating links and generalize Fox's conjecture.

Study proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.

Alternative proof of Alexander polynomial trapezoid conjecture using dimers.

New proof of trapezoidal property for Alexander polynomials of special alternating links.

Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.

Researchers describe a spectral sequence for knots in 3D space.

Study links weaving knots with polynomial coefficients and lattice numbers.

Formula for Alexander polynomial of twisted torus knots derived.

The paper shows links can be colored with fewer colors than previously thought.

Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter-Silver-Williams clarifies the relationship between the two focusing on how one transitions between Fox and Dehn colorings. I…

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and $ω$-signatures are shown to give bounds on the topologic…

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In …

New results on algebraic knots with Brieskorn polynomials.

In this paper we study embeddings of oriented connected closed surfaces in $\mathbb S^3$. We define a complete invariant, the fundamental span, for such embeddings, generalizing the notion of the peripheral system of a knot group. From the fundamental span, several computable invariants are derived and employed to stud…

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2…