A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New groups from virtual link stacks distinguish Kishino knots.
New writhe polynomial for virtual links defined using weak chord index.
Study of groups for virtual trefoil and Kishino knots.
Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…
This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the pol…
A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Al…
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 22 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from …
We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the non-triviality o…
We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots…
We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual and . These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conv…