TY - JOUR
T1 - (1, 2) and weak (1, 3) homotopies on knot projections
AU - Ito, Noboru
AU - Takimura, Yusuke
PY - 2013/12
Y1 - 2013/12
N2 - In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).
AB - In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).
KW - (1, 2) homotopy
KW - flat Reidemeister move
KW - Knot projection
KW - weak (1, 3) homotopy
UR - http://www.scopus.com/inward/record.url?scp=84892413913&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84892413913&partnerID=8YFLogxK
U2 - 10.1142/S0218216513500855
DO - 10.1142/S0218216513500855
M3 - Article
AN - SCOPUS:84892413913
SN - 0218-2165
VL - 22
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 14
M1 - 1350085
ER -