8 citations
7 papers
Thirty-two equivalence relations on knot projections
Noboru Ito, Yusuke Takimura
We consider 32 homotopy classifications of knot projections (images of generic immersions from a circle into a 2-sphere). These 32 equivalence relations are obtained based on which…
On a nontrivial knot projection under (1, 3) homotopy
Noboru Ito, Yusuke Takimura
In 2001, Östlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plan…
Crosscap number and knot projections
Noboru Ito, Yusuke Takimura
We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-t…
Knot projections with reductivity two
Noboru Ito, Yusuke Takimura
Reductivity of knot projections refers to the minimum number of splices of double points needed to obtain reducible knot projections. Considering the type and method of splicing (S…
Strong and weak (1, 2, 3) homotopies on knot projections
Noboru Ito, Yusuke Takimura
A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of k…
Sub-chord diagrams of knot projections
Noboru Ito, Yusuke Takimura
A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve provides a chord diagram by associating each ch…