We study collections of planar curves that yield diagrams for all knots. In particular, we show
that a very special class called potholder curves carries all knots. This has implications for …

Using a lattice model of polymers in a tube, we define one way to characterise different
configurations of a given knot as either “local” or “non-local”, based on a standard approach …

In this paper, we consider closed surfaces which contain spatial graphs. In the case that a
closed surface is a 2-sphere, we show that the 2-sphere can be isotoped so that it intersects …

Let F be an incompressible, meridionally incompressible and not boundary-parallel surface
with boundary in the complement of an algebraic tangle (B, T). Then F separates the strings …

Knots are ubiquitous in nature and their analysis has important implications in a wide variety
of fields including fluid dynamics, material science and molecular and structural biology. In …

The trunk of a knot in S 3, defined by Makoto Ozawa, is a measure of geometric complexity
similar to the bridge number or width of a knot. We prove that for any two knots K 1 and K 2 …

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a
variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens …

It has been conjectured that the exponential growth rate of the number of lattice polygons
with knot-type $ K $ is the same as that for unknot polygons, and that the entropic critical …

In this paper, we investigate three geometrical invariants of knots, the height, the trunk and
the representativity. First, we give a counterexample for the conjecture which states that the …

In this paper, we define invariants of links in terms of colorings of link diagrams and prove
that these invariants coincide with various notions of widths of links with respect to the …