在此篇論文中,我們討論了繩節高分子基本的結構問題,我們提出使用平均多餘交叉點數來衡量一個繩節高分子, 繩節高分子被我們視為是一個大球中包裹著好幾顆的小球,小球的個數是由拓樸結構來決定,此外我們也利用拉力實驗來抽取出繩節高分子因為拓樸限制所減少的自由能. In this thesis, we will discuss the basic physics of knotted polymers including the static structures, and elasticity. The average excess crossing number defined as the average crossing number of a given knotted polymer minus the average crossing number of the corresponding trivial knot, could be used as the fundamental quantity of a knotted polymer. Starting from the average excess crossing number, we proposed a blob picture of a knotted polymer. The knotted polymer is considered to be a large sphere with closed packed small blobs inside. With this simple intuitive argument, radius of gyration for a knotted polymer in good and theta solvent were verified. The finite size scaling law of a knotted polymer for coil to globule transition is presented and verified by computer simulation. Hence, we claimed that a knotted polymer is a statistical physics system with two characteristic lengths, the radius of gyration and the intrinsic blob size. This differs from linear polymer with only one characteristic length. We also calculated the size of a knotted polymer by using Flory-like mean field theory. And it also shows consistent results with the blob picture argument. For the stretching of a knotted polymer, there exists a characteristic crossover force $f^{*}$ which determines when the topological blobs structure will be destroyed by the stretching force. From the blobs picture of a knotted polymer, the topological free energy of a knotted polymer could be extratcted from force-extension experiment.
