arXiv research

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

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…

Computations for prime knots up to 11 crossings.

The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic n…

An algorithm determines knot colorability and determinants from petal projections.

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…

Study of origamis' singularities for groups of prime-power order.

Identifies doubly slice genera for 2909 prime knots with up to 12 crossings.

This paper finds all prime alternating knots with minimal warping degree two.

Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element tha…

Cyclic covers of knots uniquely determine the original knot.

Study grid homology of diagonal knots, finding key terms related to prime factors and decompositions.

We associate an open book with any connected plane checkerboard graph, thus providing a common extension of the classes of prime positive braid links and positive tree-like Hopf plumbings. As an application, we prove that the link type of a prime positive braid closure is determined by the linking graph associated with…

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complemen…

Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a m…

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. …

The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that non…

Study identifies prime strongly positive amphicheiral knots with double symmetry.

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …

The Burau representation of 3-strand braid group modulo p is determined and shown to be faithful for small p.

For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of whether a given (prime) knot is a 2-bridge knot remains a nontrivial exercise, and …

Let $K$ be a prime knot in $S^3$ and $G(K)=π_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then $640,800$ shou…

We determine prime amphicheiral links with at least 2 components and up to 11 crossings. There are 27 such links. We check also special amphicheiralities. Most of prime links with up to 11 crossings are detected not to be amphicheiral by a condition on the Jones polynomial. For the rest links, we applied conditions fro…

Study theta-curves on torus in 3-sphere, classifying them.

The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, $K$ and $K^{\prime}$, intersecting at two points transversely. Each of $K$ and $K^{\prime}$ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along…

This paper solves the structure of link concordance groups, proving they are infinitely generated.

In this paper we describe methods for computing rack and quandle cohomology. We illustrate these methods by completely determining the cohomology of prime dihedral quandles.

Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 1…

In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer $n$ such that a knot or link can be represented on an $n$-mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest…

The paper identifies knots in specific lens spaces based on their complements.

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $λ$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $λ$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In …

The Kauffman-Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. In this way…

The paper determines modular cohomotopy groups up to extensions using classical and unstable homotopy methods.

Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decompositio…

We show that, for any prime p, a knot K in the 3-sphere is determined by its p-fold cyclic unbranched covering. We also investigate when the m-fold cyclic unbranched covering of a knot coincides with the n-fold cyclic unbranched covering of another knot, for different coprime integers m and n.

We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use $m$ twists, and, rather than bisecting ($n = 2$), cut the strip into $n$ sections. We call the resulting collection of t…

Delta-unlinking number measures how to unlink algebraically split links.

We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knot…

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

We will show that if $K$ is a knot of prime period $p>2$ and whose Alexander polynomial $Δ_K(t)$ is monic and of degree $p-1$, then $Δ_K(t)$ is uniquely determined only by $p$.

The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. The…

The paper determines the maximal order of translation groups in abelian differentials for various genera.

The study establishes a link between fibered links and their concordance invariants.

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knots of type (2, p), modulo p. This expression is t + 2 l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations…