A Survey of Knot Concordance (Low-Dimensional Topology of Tomorrow)
Andrius Tamulis · Kyoto University Research Information Repository (Kyoto University) · 2002
At the opening of the conference, our host, Hitoshi Murakami, charged the participants to consider the past, present, and future of low-dimensional topology.Itherefore arrange this paper to follow that format.1The Past Knot ConcordanceAknot is an embedding of an oriented $S^{1}$ in an oriented $S^{3}$ .We work here up to diffe0- morphism or piecewise-linear locally flat homomorphism.Aknot K is concordant to a knot J(K $\simeq J)$ if there exists amanifold pair W, consisting of an annulus embedded in athree sphere cross an interval, W $\cong(S^{1}\cross I)\mathrm{c}arrow(S^{3}\cross I)$ , such that $\partial W\cong K\cup-J$ .Here, -J means the knot J with the orientation reversed on both the ambient $S^{3}$ and the embedded $S^{1}$ .Concordance is an equivalence relation, and it respects connected sums of knots; i.e. if K $\simeq K'$ and J $\simeq J'$ then $K\# J$ $\simeq K'\# J'$ .Thus we can form agroup $C_{1}$ out of