It is well known that the total torsion of a closed spherical curve is zero. Furthermore, if the
total torsion of any closed curve on the surface is zero, then it is part of a plane or a sphere …

Every knot projection is simplified to the trivial spherical curve not increasing double points
by using deformations of types 1, 2, and 3 which are analogies of Reidemeister moves of …

Although it is known that the dimension of the Vassiliev invariants of degree three of long
virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been …

For a pair of spherical curves P and P′, we introduce a quantity called the stable double
point number of P and P′, denoted sd (P, P′). A trivial spherical curve is a spherical curve …

Let L be a fixed link. Given a link diagram D, is there a sequence of crossing exchanges and
smoothings on D that yields a diagram of L? We approach this problem from the …

Abstract In [1], a deformation of spherical curves called deformation type α was introduced.
Then, it was showed that if two spherical curves P and P′ are equivalent under the relation …

The space of Gauss diagram formulas that are knot invariants is introduced by Goussarov–
Polyak–Viro in 2000; it is extended to nanophrases by Gibson–Ito in 2011. However, known …

A generic immersion of a circle into a $2 $-sphere is often studied as a projection of a knot; it
is called a knot projection. A chord diagram is a configuration of paired points on a circle; …

A plane graph $ H $ is a {\em plane minor} of a plane graph $ G $ if there is a sequence of
vertex and edge deletions, and edge contractions performed on the plane, that takes $ G …