Distributive Groupoids in Knot Theory

Abstract

PACS. 71.55 -i -Impurity and defect levels. PACS. 72.20Ht -High field and nonlinear effects.PACS. 72.40 + w -Photoconduction and photovoltaic effects: photodielectric effects.Abstract. -The ionisation of deep impurity centres in germanium has been observed with radiation in the terahertz range where the photon energy is much less than the binding energy of the impurities. It is shown that for not too high radiation intensities the ionisation is caused by the Poole-Frenkel effect. As in the well-known case of d.c. fields, the electric field of the highfrequency radiation lowers the Coulomb potential barrier and enhances the thermal emission of carriers.

Twisted quandle homology theory and cocycle knot invariants

“…Racks and quandles have been studied in, for example, [1,13,20,21,23]. The axioms for a quandle correspond respectively to the Reidemeister moves of type I, II, and III (see [13,21], for example).…”

Equivalence classes of colorings

“…In order for this notion to be topological we require a special kind of permutation which we call a Coloring Automorphism. These are permutations which comply with the coloring operation, a * b := 2b − a in a pre-assigned modulus m. This operation generalizes to the quandle operation, generalizing also the notion of coloring ( [7,13]). In the particular instance a * b = 2b − a we are dealing with the so-called dihedral quandles, one per integer modulus m.…”

“…In the context of quandles this work has to do with homomorphisms from the fundamental quandle of the knot to the dihedral quandles ( [7,13]). We organize these homomorphisms into equivalence classes.…”

Triple point numbers of surface-links and symmetric quandle cocycle invariants

“…A quandle (see Fenn and Rourke [3], Joyce [5] or Matveev [10]) is a set X with a binary operation .x; y/ 7 ! x y such that (i) for any x 2 X , it holds that x x D x , (ii) for any x; y 2 X , there exists a unique z 2 X such that z y D x , and (iii) for any x; y; z 2 X , it holds that .x y / z D .x z / .y z / .…”