11 citations · 30 across the 12 of their papers we have counts for
18 papers
Knot-theoretic flocks
Maciej Niebrzydowski, Agata Pilitowska, Anna Zamojska-Dzienio
We characterize the para-associative ternary quasigroups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We…
Layers of knot region colorings and higher differentials
Maciej Niebrzydowski
We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differen…
On a homology of ternary groups with applications to knot theory
Maciej Niebrzydowski
We define a homology for ternary groups using both associativity and skew elements. We describe the odd-even construction which yields many examples of ternary groups. We define th…
Knot-theoretic ternary groups
Maciej Niebrzydowski, Agata Pilitowska, Anna Zamojska-Dzienio
We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special att…
Ternary quasigroups in knot theory
Maciej Niebrzydowski
We show that some ternary quasigroups appear naturally as invariants of classical links and links on surfaces. We also note how to obtain from them invariants of Yoshikawa moves. I…
Homology of ternary algebras yielding invariants of knots and knotted surfaces
Maciej Niebrzydowski
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satis…
Categories of diagrams with irreversible moves
Maciej Niebrzydowski
We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition, regarding the existence of certain…
On some ternary operations in knot theory
Maciej Niebrzydowski
We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitut…
Entropic magmas, their homology, and related invariants of links and graphs
Maciej Niebrzydowski, Józef H. Przytycki
We define link and graph invariants from entropic magmas modeling them on the Kauffman bracket and Tutte polynomial. We define the homology of entropic magmas. We also consider gro…
The second quandle homology of the Takasaki quandle of an odd abelian group is an exterior square of the group
Maciej Niebrzydowski, Jozef H. Przytycki
We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology of the Takasaki quandle of G to the exterior square of G. In parti…
Coloring invariants of spatial graphs
Maciej Niebrzydowski
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the grap…
Homology operations on homology of quandles
Maciej Niebrzydowski, Jozef H. Przytycki
We consider various homological operations on homology of quandles. We introduce the notion of quandle partial derivatives, and extreme chains on which appropriate partial derivati…
Three geometric applications of quandle homology
Maciej Niebrzydowski
In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distanc…
The quandle of the trefoil knot as the Dehn quandle of the torus
Maciej Niebrzydowski, Jozef H. Przytycki
We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space . The last qua…
Biquandle longitude invariant of long virtual knots
Maciej Niebrzydowski
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subt…
Homology of dihedral quandles
Maciej Niebrzydowski, Jozef H. Przytycki
We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H_3^R(R_p) = Z \oplus Z_p for p odd prime. We also show that H_…
On colored quandle longitudes and its applications to tangle embeddings and virtual knots
Maciej Niebrzydowski
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-clas…
Burnside kei
Maciej Niebrzydowski, Jozef H. Przytycki
This paper is motivated by a general question: for which values of k and n is the universal Burnside kei of k generators and Kei "exponent" n, , finite? It is known (s…