Physical knots and links are one-dimensional submanifolds of R with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$ . $\mathcal {C}_t = Sh(M, \Lambda _t)$ the differential graded derived category constructible sheaves on $M$ with support at infinity contained _t$ We prove if embeds into Liouville hypersurfaces, the...

By associating a collection of knots and links to a graph in threedimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigid-vertex isotopy. Rigid-vertex graphs are a category mixing topological flexibility with mechanical rigidity. Both categories constitute steps toward models for chemical and biological ne...

To date, many ways for encoding links have been discovered, see, e.g., [1, 5], and [3]. One of them is the encoding by closures of braids (Alexander’s theorem, see, e.g., [6]). Thus, it is important to find a ‘good’ class of braids that encodes all link isotopy types. In the present paper, we prove that all link isotopy classes can be encoded by a very small class of so-called quasitoric braids...

A complex projective (or CP ) structure on a compact surface S is an atlas of charts with values in CP and Möbius transition functions. Let P.S/ denote the space of (isotopy classes of) marked complex projective structures on S , and let T .S/ be the Teichmüller space of (isotopy classes of) marked complex structures on S . Because Möbius maps are holomorphic, there is a forgetful projection W ...

There is a natural way to define an isomorphism between the group of transformations of isotopy classes of rational tangles and the modular group. This isomorphism allows to give a simple proof of the Conway theorem, stating the one-to-one correspondence between isotopy classes of rational tangles and rational numbers. Two other simple ways to define this isomorphism, one of which suggested by ...

Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are i...