arXiv research

New invariant for virtual n-links defined and studied.

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to e…

Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group $\mathscr{VC}$. It is shown that for every concordance cla…

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hype…

We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the non-triviality o…

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fi…

Study shows patterns in Stein fillability of trefoil surgeries.

Wilson-loop averages in Chern-Simons theory (HOMFLY polynomials) can be evaluated in different ways -- the most difficult, but most interesting of them is the hypercube calculus, the only one applicable to virtual knots and used also for categorification (higher-dimensional extension) of the theory. We continue the stu…

Proved colored HOMFLY-PT polynomials for specific knots.

The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is…

We prove that the reduced 2-coloured Khovanov homology detects the trefoil, using a spectral sequence to knot Floer homology.

Study cobordism distances between 3-braid links and trefoil knots.

Efficient cobordisms show minimal signature values on certain links.

With the aid of a computer, we provide a motion picture of the twist-spun trefoil which exhibits the periodicity well.

2-twist trefoil has 6 crossings, proving non-trivial knotted surface.

Smooth knots with odd Conway polynomial terms have inscribed trefoils.

We give a criterion for an open book to contain an n-times iterated Hopf plumbing summand. As an application, we show that fibre surfaces of positive braid knots admit a trefoil plumbing structure.

This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots …

New knot quandle structure for twist-spun trefoils discovered.

Numerical computations suggest that each point on a certain optimized shape called the ideal trefoil is in contact with two other points. We consider sequences of such contact points, such that each point is in contact with its predecessor and call it a billiard. Our numerics suggest that a particular billiard on the i…

The paper studies knots in modular flows using self-covers.

New computations show sl(N) homology is related to SU(N) representations of knots.

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\Z \oplus \Z$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fund…

New knots are found to be non-simple in Legendrian contact geometry.

Khovanov homology detects specific links.

We show that the knot quandle of the $3$-, $4$-, or $5$-twist-spun trefoil is isomorphic to a quandle related to the $16$-, $24$-, or $600$-cell respectively. We further show that the cardinality of the knot quandle of the $m$-twist-spun trefoil is finite if and only if $1 \leq m \leq 5$. This phenomenon is attributabl…

A filling Dehn surface in a $3$-manifold $M$ is a generically immersed surface in $M$ that induces a cellular decomposition of $M$. Given a tame link $L$ in $M$ there is a filling Dehn sphere of $M$ that "trivializes" (\emph{diametrically splits}) it. This allows to construct filling Dehn surfaces in the coverings of $…

Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A …

We prove all knots can be transformed into a trefoil using special diagrams.

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

Minimal area of spun trefoil knot is found in 4D cubical space.

We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini.

New method detects and compares folding pathways of knotted proteins.

A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is not isotopic to its orientation reverse. We sketch the proof the classification …

We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 1…

We prove that if the lens space $L(n, 1)$ is obtained by a surgery along a knot in the lens space $L(3,1)$ that is distance one from the meridional slope, then $n$ is in $\{-6, \pm 1, \pm 2, 3, 4, 7\}$. This result yields a classification of the coherent and non-coherent band surgeries from the trefoil to $T(2, n)$ tor…

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spe…

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant appr…

Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …

Mathematical pipeline identifies structural homology of knotted proteins.

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic…

We explore free knot diagrams, which are projections of knots into the plane which don't record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple fami…

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $γ: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot of type $K$ in space. We prove that there exists a sequence of numbers $0\leq t_1 < t…

This paper gives mathematical models for flat knotted ribbons, and makes specific conjectures for the least length of ribbon (for a given width) needed to tie the trefoil knot and the figure eight knot. The first conjecture states that (for width one) the least length of ribbon needed to tie an open-ended trefoil knot …

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…

The complete classification of representations of the Trefoil knot group G in S^{3} and SL(2,R), their affine deformations, and some geometric interpretations of the results, are given. Among other results, we also obtain the classification up to conjugacy of the non cyclic groups of affine Euclidean isometries generat…

The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to q…