Journal of Knot Theory and Its Ramifications

It has been conjectured that for knots K and K′ in S³, w(K # K′) = w(K) + w(K′) - 2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots for which they conjectured that where Bⁿ is a bridge number n knot. We show that for n > 2 none of the knots in produces such counterexamples.

We study knots in 𝕊³ obtained by the intersection of a minimal surface in ℝ⁴ with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

Let M be a compact orientable 3-manifold, and A an essential annulus which cuts M into two 3-manifolds M₁ and M₂. We denote by g(M) the Heegaard genus of M. In this paper, we will give a lower bound to g(M) when M_i contains no bounded essential surfaces with large Euler's characteristic, where all boundary components of the essential surfaces lie in A.

We compare the path-width of a planar graph and the bridge number of the corresponding link. Is there a relationship between these two numbers? This study is an attempt to find a graph theoretical method of finding the bridge number of a link.

A Chebyshev knot is a knot which admits a parametrization of the form x(t) = T_a(t); y(t) = T_b(t); z(t) = T_c(t + ϕ), where a, b, c integers, T_n(t) is the Chebyshev polynomial of degree n, and φ ∈ R. Chebyshev knots are non-compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with φ = 0. We also show that every knot is a Chebyshev knot.

The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers.

We show that the bridge number of a tunnel number t knot in S³ with respect to an unknotted genus t surface is bounded below by a function of the distance of the Heegaard splitting induced by the t tunnels. It follows that for any natural number n, there is a tunnel number one knot in S³ that is not (1, n).

Let K be a non-trivial knot in the 3-sphere with a lens space surgery and L(p, q) a lens space obtained by a Dehn surgery on K. We study a relationship between the order p of the fundamental group of L(p, q) and the Seifert genus g of K. Considering certain infinite families of knots with lens space surgeries, the following estimation is suggested as a conjecture: except for (g, p) = (5, 19).

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we analyze the trivializing number of a pseudodiagram, i.e. the minimum number of crossings that must be resolved to produce the unknot. We consider how much crossing information is needed in a virtual pseudodiagram to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.