Abstract
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 knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first nontrivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e. Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.
Original language | English |
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Article number | 1550069 |
Journal | International Journal of Mathematics |
Volume | 26 |
Issue number | 9 |
DOIs | |
Publication status | Published - 2015 Aug 29 |
Externally published | Yes |
Keywords
- Generic spherical curves
- homotopy
- knot projections
- Reidemeister moves
ASJC Scopus subject areas
- Mathematics(all)