We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.
%0 Journal Article
%1 Tanaka2007a
%A Tanaka, Kokoro
%D 2007
%J Topology and its Applications
%K algebra knot-theory quandle topology
%N 15
%P 2757--2763
%R 10.1016/j.topol.2007.05.010
%T Inequivalent surface-knots with the same knot quandle
%U http://dx.doi.org/10.1016/j.topol.2007.05.010
%V 154
%X We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.
@article{Tanaka2007a,
abstract = {We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.
},
added-at = {2009-05-26T15:29:21.000+0200},
author = {Tanaka, Kokoro},
biburl = {https://www.bibsonomy.org/bibtex/2ead1742bc1da0baf4975e425a11cce41/njj},
doi = {10.1016/j.topol.2007.05.010},
interhash = {42b5d39cdc60f693cbf2a65a04f63b15},
intrahash = {ead1742bc1da0baf4975e425a11cce41},
issn = {0166-8641},
journal = {Topology and its Applications},
keywords = {algebra knot-theory quandle topology},
number = 15,
pages = {2757--2763},
timestamp = {2009-05-26T15:29:21.000+0200},
title = {Inequivalent surface-knots with the same knot quandle},
url = {http://dx.doi.org/10.1016/j.topol.2007.05.010},
volume = 154,
year = 2007
}