Low-dimensional topology; knot concordance and geography

三维空间中的纽结称为协调纽结，如果它们共同约束三维空间与闭区间的乘积中的嵌入环。 和谐类的集合形成了一个阿贝尔群，本项目研究这个和谐群的结构。 该项目主要有三个方面：（1）更好地理解已知的和谐不变量并发现新的不变量;（2）应用这些不变量对重要的纽结族进行分类;（3）利用这些结果揭示和谐群的潜在结构。经典物理学的视角是三维的，关注我们生活的三个空间维度。 现代物理学有一个更高维度的视角，例如包括时间来创建宇宙的四维模型。 在20世纪60年代，人们认识到四维空间可以通过将更简单的部分粘合在一起来构建;这些部分粘合在一起的方式可以用结来表示，由此产生的空间的复杂性反映在发生的结的复杂性中。 这个建议的重点是从四维的角度来理解打结。 特别是，新的工具将被开发，这些将被应用于完成我们的低交叉数节的四维分析。

Knots in 3-space are called concordant if they cobound an embedded annulus in the product of 3-space with a closed interval. The set of concordance classes forms an abelian group; this project investigates the structure of this concordance group. There are three major facets to the project: (1) Developing a better understanding of known concordance invariants and discovering new invariants, (2) Applying these invariants to classify important families of knots, and (3) Using these results to unravel the underlying structure of the concordance group.The perspective of classical physics is three-dimensional, focusing on the three spatial dimensions in which we live. Modern physics has a higher-dimensional perspective, for instance including time to create a four-dimensional model of the universe. In the 1960s it was recognized that four-dimensional spaces can be built by gluing together simpler pieces; the way these pieces are glued together can be represented by knots, with the complexity of the resulting space reflected in the complexity of the knots that occur. This proposal is focused on understanding knotting from a four-dimensional perspective. In particular, new tools will be developed, and these will be applied to complete our four-dimensional analysis of low- crossing number knots.