Energy of a knot: variational principles; Mm-energy

Generalization of Polynomial Invariants and Holographic Principle for Knots and Links

We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for polynomial invariants of torus knots T(2m+1,2) and torus links L(2m,2). As an example of the application of this principle, we derive the Jones skein relation and its generalization with the help of some variants of (q,p)-numbers, related with (q,p)-deformed bosonic oscillators.

Crossing indices, traits and the principle of indistinguishability

A (weak chord) index is a function on the crossings of knot diagrams such that: 1) the index of a crossing does not change under Reidemeister moves; 2) crossings which can be paired by a second Reidemeister move have the same index. We show that one can omit the second condition in the case of the universal index. As a consequence, we get the following principle of indistinguishability for classical knots: crossings of the same sign in a classical knot diagram can not be distinguished by any inherent property.

$h$-原理：具有指定曲率的光滑曲线与纽结

$h$-Principles for smooth curves and knots with prescribed curvature

本文证明了具有指定曲率的光滑曲线满足$C^1$-稠密$h$-原理，从而在$R^3$中每个同痕类里都存在指定曲率的$C^\infty$纽结。

One-Term Parity Bracket For Braids

In previous papers, the author realized the following principle for many knot theories: if a knot diagram is complicated enough then it reproduces itself, i.e., is a subdiagram of any other diagram equivalent to it. This principle is realized by diagram-valued invariants [ ] of knots such that [K]=K. It turns out that in the case of free braids, the same principle can be realized an unexpectedly easy way by a one-term invariant formula.

Classification of Engel Knots

For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.

Lipshitz–Ozsváth–Thurston对应关系的不变分裂原理

Invariant splitting principles for the Lipshitz--Ozsváth--Thurston correspondence

该研究证明Lipshitz-Ozsváth-Thurston对应关系的不变分裂原理，给出从伴随纽结对合Floer同调计算卫星同调的新方法，还构造了无限多个一次稳定化后仍奇异的可缩4-流形对。