arXiv research

The paper creates knot invariants using free groups.

We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.

Study on random knot diagrams and their probability of forming specific knots.

A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genu…

In [D.A. Fedoseev, V.O. Manturov, A sliceness criterion for odd free knots,arXiv:1707.04923], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper we give a similar criterion for stably odd free knots. Some additional results on knot sliceness and cobordism are g…

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order >2.

Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that is irreducibly odd, then it is minimal with respect to the number of classical …

Study knot invariants using automorphism groups of free nilpotent groups.

New knot groups found to be bi-orderable using pretzel knots.

We determine the structure of the circular handle decompositions of the family of free genus one knots. Namely, if k is a free genus one knot, then the handle number h(k)= 0, 1 or 2, and, if k is not fibered (that is, if h(k)>0), then k is almost fibered. For this, we develop practical techniques to construct circular …

New proof shows knot quandles of ribbon knots are distinct.

The main goal of this paper is to prove that for odd free knots - that is free knots with all odd crossings - the problem of sliceness (the existence of a spanning disc) has an explicit answer based on the pairing of the knot diagram chords.

Universal Gaussian parity proven for 2D knots.

Classifies knots that bound equivariant surfaces with free symmetries.

New method computes knot invariants using free group automorphisms.

We show that free genus of knots is additive under connected sum.

Conditions for integer signatures of high-dimensional knots.

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-t…

This paper studies periodic and free periodic knots in alternating projections.

Paper develops knot invariants for long knots in a torus.

We study the structure underlying Ng's conjecture, which relates the degree $0$ abelian knot contact homology of a knot $K$ to the coordinate ring of the $SL_2(\mathbf{C})$-character variety $X(Σ_2 K)$ of the $2$-fold branched cover of the $3$-sphere branched along $K$. Our approach is based on the study of (meridional…

Study on knotting in very long polymer chains, finding Poisson distribution for prime knot types.

In this paper, we introduce a notion called n/k-free tangle and study the degeneration ratio of tunnel numbers of knots.

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by term…

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial fre…

A Seifert surface F for a knot K is disk decomposable if there is a taut sutured manifold heirarchy for the complement of F, whose decomposing surfaces are all disks. It follows that F has minimal genus for the knot K, and has handlebody complement, i.e., F is free. We show that these necessary conditions for disk deco…

Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…

We show that every $\mathbb{Z}$-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the …

Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at …

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev, who thought all free knots to be trivial). As it turned …

By using parity arguments we prove that free knots are, generally, not invertible.

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

Study on periodic knots, proving limitations on their Alexander polynomials.

We construct a simple invariant of free link valued in a certain group by using parity.

We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…

The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.

It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot $5_2$, which is the $(-2)$-twist knot, by Naylor and Rolfsen.

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.

We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.

Model proteins with bonds using Kauffman bracket skein module.

We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links and looped graphs.

Method determines SL(2,C) character variety for Montesinos knots.

The paper shows geometric realisation over specific groups and knots.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$…