ASurvey of Knot Concordance

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 1.1 Knot Concordance Aknot 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 o