arXiv research

Minimal generating sets of Reidemeister moves identified and classified.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

Minimal moves for surfaces in 4D identified.

Minimal sets of moves for rotational Reidemeister diagrams are identified.

Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.

Paper solves the minimal generating set problem for singular Reidemeister moves.

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof al…

Polyak proved that the set $\{\Omega1a,\Omega1b,\Omega2a,\Omega3a\}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $…

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introd…

Yoshikawa moves were introduced at least quarter-century ago and are still actively used by researchers. For any marked graph diagram we will define its twisted diagram and its mirror cut surface. By using a surface-link group of a mirror cut surface of a twisted diagram we will prove the independence of Yoshikawa eigh…

Enumerates knots up to five crossings and describes moves between them.

In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By intro…

Study on minimal surfaces in Heisenberg group with duality formula.

Extends symmetry and rigidity to surfaces with soap film-like singularities.

New formulas limit minimal submanifolds' area in curved spaces.

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

Let $D(p,q)$ be the usual knot diagram of the $(p,q)$-torus knot, that is, $D(p,q)$ is the closure of the $p$-braid $(σ_1^{-1} σ_2^{-1}... σ_{p-1}^{-1})^q$. As is well-known, $D(p,q)$ and $D(q,p)$ represent the same knot. It is shown that $D(n+1,n)$ can be deformed to $D(n,n+1)$ by a sequence of $\{(n-1)n(2n-1)/6 \} + …

The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.

This study simplifies verification of invariants in oriented virtual knots.

We show that the 14 graphs obtained by $\nabla\mathrm{Y}$ moves on K_7 constitute a complete list of the minor minimal intrinsically knotted graphs on 21 edges. We also present evidence in support of a conjecture that the 20 graph Heawood family, obtained by a combination of $\nabla\mathrm{Y}$ and $\mathrm{Y}\nabla$ mo…

Arnold introduced invariants $J^+$, $J^-$ and $St$ for generic planar curves. It is known that both $J^+ /2 + St$ and $J^- /2 + St$ are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting.…

Neural nets solve braid untangling up to length 20.

Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.

We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot …

In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on $K_7$, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not $2$--apex. As a corollary, this gives a new proof that the 14 graphs obtained by triangle-Y moves on $K_7$ are t…

The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.

This paper disproves a conjecture about knot projections under specific homotopy conditions.

New proof of Lie-Tresse theorem with computational advantages.

In mathematics, a knot is a single strand of string crossed over itself any number of times, and connected at the ends. The Reidemeister Moves have been proven to be the three core moves necessary to fully untangle a knot. Some knots can be untangled to a loop (the unknot), while others are fundamentally knotted. We de…

This note has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diag…

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classi…

Study shows fibred knots can't be untied with specific moves.

New method estimates velocity fields for minimizing $f$-divergences without overfitting.

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

Discrete knot theory models use lattice-filtered graphs to detect merging knot components.

We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.

Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give a simpler proof, utilizing a normal form for surfaces whose boundary is containe…

This is the first of three papers that refine and extend portions of our earlier preprint, "Depth of a knot tunnel." Together, they rework the entire preprint. H. Goda, M. Scharlemann, and A. Thompson described a general construction of all tunnels of all tunnel number 1 knots using "tunnel moves". We apply the theory …

Extends knot polynomial to knotted 4-valent graphs.

New minimal surfaces found from vortex crystals.

In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is "almost graph…

Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space $\mathbb{A}^3$. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…

Unified invariant for immersed surface-links using biquandle cocycles.

The paper defines new homotopy relations on knot projections and classifies certain knot types.

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 …