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Full-Text Articles in Physical Sciences and Mathematics
The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell
The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell
Alissa Crans
Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbid- den number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.
Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson
Polynomial Knot And Link Invariants From The Virtual Biquandle, Alissa S. Crans, Allison Henrich, Sam Nelson
Alissa Crans
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 …