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On some ternary operations in knot theory

100%

Niebrzydowski M.

Fundamenta Mathematicae

2014

tom 225

nr 1

259-276

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 substitutions in the algebras that we assign to diagrams, we obtain the relations from the knot group, and from the core group. Using the ternary operator approach, we generalize the Dehn presentation of the knot group to extra loops, and a similar presentation for the core group to the variety of Moufang loops.

Burnside kei

63%

Niebrzydowski M., Przytycki J.

Fundamenta Mathematicae

2006

tom 190

nr 1

211-229

This paper is motivated by a general question: for which values of k and n is the universal Burnside kei Q̅(k,n) finite? It is known (starting from the work of M. Takasaki (1942)) that Q̅(2,n) is isomorphic to the dihedral quandle Zₙ and Q̅(3,3) is isomorphic to Z₃ ⊕ Z₃. In this paper, we give a description of the algebraic structure for Burnside keis Q̅(4,3) and Q̅(3,4). We also investigate some properties of arbitrary quandles satisfying the universal Burnside relation a = ⋯ a∗b∗ ⋯ ∗a∗b. Invariants of links related to the Burnside kei Q̅(k,n) are invariant under n-moves.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Niebrzydowski M.

1 Przytycki J.

1 2014

1 2006

Wyniki wyszukiwania

On some ternary operations in knot theory

Burnside kei