Related papers: The first rational Chebyshev knots

In this paper, we discuss the region unknotting number of different classes of 2-bridge knots. In particular, we provide region unknotting number for the classes of $2$-bridge knots whose Conway notation is $C(m,\ n), C(m,\ 2,\ m),$ $ C(m,\…

From Euclid's fundamental formula for the Pythagorean triples we define the rational triples relating certain congruent numbers by an identity and explore their relationships. We introduce two geometric methods relating the congruent number…

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and…

We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most…

For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these…

The values that the first two Vassiliev invariants take on prime knots with up to fourteen crossings are considered. This leads to interesting fish-like graphs. Several results about the values taken on torus knots are proved.

An explicit formula for the $A$-polynomial of the knot having Conway's notation $C(2n,4)$ is computed up to repeated factors. Our polynomial contains exactly the same irreducible factors as the $A$-polynomial defined in~\cite{CCGLS1}.

We investigate the behavior of the SL(2,C) Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the…

In the 1980's Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni,…

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…

We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…

We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with…

We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which…

We determine the prime strongly positive amphicheiral knots up to 16 crossings and show that a large fraction of them admit knot diagrams with a double symmetry (rotational symmetry for strongly positive amphicheirality and an additional…

Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If…

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal…

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average…

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…

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms…

We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.