Abstract
Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.
| Original language | English |
|---|---|
| Pages (from-to) | 341-365 |
| Number of pages | 25 |
| Journal | Mathematische Zeitschrift |
| Volume | 274 |
| Issue number | 1-2 |
| DOIs | |
| Publication status | Published - Jun 1 2013 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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Quandle homotopy invariants of knotted surfaces. / Nosaka, Takefumi.
In: Mathematische Zeitschrift, Vol. 274, No. 1-2, 01.06.2013, p. 341-365.Research output: Contribution to journal › Article
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TY - JOUR
T1 - Quandle homotopy invariants of knotted surfaces
AU - Nosaka, Takefumi
PY - 2013/6/1
Y1 - 2013/6/1
N2 - Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.
AB - Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.
UR - http://www.scopus.com/inward/record.url?scp=84877725129&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84877725129&partnerID=8YFLogxK
U2 - 10.1007/s00209-012-1073-1
DO - 10.1007/s00209-012-1073-1
M3 - Article
AN - SCOPUS:84877725129
VL - 274
SP - 341
EP - 365
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 0025-5874
IS - 1-2
ER -