Polynomial Invariant of Molecular Circuit Topology

The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.

Polynomial invariants of singular knots and links

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.

On an integrability criterion for a family of cubic oscillators

<abstract><p>In this work we consider a family of cubic, with respect to the first derivative, nonlinear oscillators. We obtain the equivalence criterion for this family of equations and a non-canonical form of Ince Ⅶ equation, where as equivalence transformations we use generalized nonlocal transformations. As a result, we construct two integrable subfamilies of the considered family of equations. We also demonstrate that each member of these two subfamilies possesses an autonomous parametric first integral. Furthermore, we show that generalized nonlocal transformations preserve autonomous invariant curves for the equations from the studied family. As a consequence, we demonstrate that each member of these integrable subfamilies has two autonomous invariant curves, that correspond to irreducible polynomial invariant curves of the considered non-canonical form of Ince Ⅶ equation. We illustrate our results by two examples: An integrable cubic oscillator and a particular case of the Liénard (4, 9) equation.</p></abstract>

Quandle Module Quivers

We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant which specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.

Some new examples of links with the same polynomials

We call a link (knot) [Formula: see text] to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to [Formula: see text] but share the same Jones (respectively, Homfly) polynomial as [Formula: see text]. We reconstruct Kanenobu’s knot [Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158–162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] ([Formula: see text] is the mirror image of [Formula: see text]) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homfly undetectable.