143 research outputs found
On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which
strand goes over/under at certain crossings may be missing. Pseudoknots are
equivalence classes of pseudodiagrams, with equivalence defined by a class of
Reidemeister-type moves. In this paper, we introduce two natural extensions of
classical knot colorability to this broader class of knot-like objects. We use
these definitions to define the determinant of a pseudoknot (i.e. the
pseudodeterminant) that agrees with the classical determinant for classical
knots. Moreover, we extend Conway notation to pseudoknots to facilitate the
investigation of families of pseudoknots and links. The general formulae for
pseudodeterminants of pseudoknot families may then be used as a criterion for
p-colorability of pseudoknots.Comment: 22 pages, 24 figure
Semiquandles and flat virtual knots
We introduce an algebraic structure we call semiquandles whose axioms are
derived from flat Reidemeister moves. Finite semiquandles have associated
counting invariants and enhanced invariants defined for flat virtual knots and
links. We also introduce singular semiquandles and virtual singular
semiquandles which define invariants of flat singular virtual knots and links.
As an application, we use semiquandle invariants to compare two Vassiliev
invariants.Comment: 14 page
Polynomial knot and link invariants from the virtual biquandle
The Alexander biquandle of a virtual knot or link is a module over a
2-variable Laurent polynomial ring which is an invariant of virtual knots and
links. The elementary ideals of this module are then invariants of virtual
isotopy which determine both the generalized Alexander polynomial (also known
as the Sawollek polynomial) for virtual knots and the classical Alexander
polynomial for classical knots. For a fixed monomial ordering , the
Gr\"obner bases for these ideals are computable, comparable invariants which
fully determine the elementary ideals and which generalize and unify the
classical and generalized Alexander polynomials. We provide examples to
illustrate the usefulness of these invariants and propose questions for future
work.Comment: 12 pages; version 3 includes corrected figure
A Midsummer Knot's Dream
In this paper, we introduce playing games on shadows of knots. We demonstrate
two novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We
also discuss winning strategies for these games on certain families of knot
shadows. Finally, we suggest variations of these games for further study.Comment: 11 pages, 8 figures. To appear, College Mathematics Journal
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