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 |
|---|---|
| Article number | 1550069 |
| Journal | International Journal of Mathematics |
| Volume | 26 |
| Issue number | 9 |
| DOIs | |
| Publication status | Published - 2015 Aug 29 |
| Externally published | Yes |
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Keywords
- Generic spherical curves
- homotopy
- knot projections
- Reidemeister moves
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Strong and weak (1, 2, 3) homotopies on knot projections. / Ito, Noboru; Takimura, Yusuke.
In: International Journal of Mathematics, Vol. 26, No. 9, 1550069, 29.08.2015.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Strong and weak (1, 2, 3) homotopies on knot projections
AU - Ito, Noboru
AU - Takimura, Yusuke
PY - 2015/8/29
Y1 - 2015/8/29
N2 - 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.
AB - 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.
KW - Generic spherical curves
KW - homotopy
KW - knot projections
KW - Reidemeister moves
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UR - http://www.scopus.com/inward/citedby.url?scp=84940374938&partnerID=8YFLogxK
U2 - 10.1142/S0129167X1550069X
DO - 10.1142/S0129167X1550069X
M3 - Article
VL - 26
JO - International Journal of Mathematics
JF - International Journal of Mathematics
SN - 0129-167X
IS - 9
M1 - 1550069
ER -