Minimal Diagrams of Free Knots
نویسندگان
چکیده
An irreducibly odd graph is a graph such that each vertex has odd degree and for every pair of vertices, a third vertex in the graph is adjacent to exactly one of the pair. This family of graphs was introduced recently by Manturov [1] in relation to free knots. In this paper, we show that every graph is the induced subgraph of an irreducibly odd graph. Furthermore, we prove that irreducibly odd graphs must contain a particular minor called the triskelion.
منابع مشابه
A Partial Ordering of Knots Through Diagrammatic Unknotting
In this paper we define a partial order on the set of all knots and links using a special property derived from their minimal diagrams. A knot or link K′ is called a predecessor of a knot or link K if Cr(K′) < Cr(K) and a diagram of K′ can be obtained from a minimal diagram D of K by a single crossing change. In such a case we say that K′ < K. We investigate the sets of knots that can be obtain...
متن کاملMinimal diagrams of classical knots
We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms and knots.
متن کاملQuasi-alternating Links and Odd Homology: Computations and Conjectures
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples ...
متن کاملOn the unknotting number of minimal diagrams
Answering negatively a question of Bleiler, we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent.
متن کاملOn Some Restrictions to the Values of the Jones Polynomial
We prove a relation in the algebra of the Jones Vassiliev invariants, giving a new restriction to the values of the Jones polynomial on knots, and the non-existence of another family of such relations. We prove, that Jones polynomials of positive knots have non-negative minimal degree and extend this result to k-almost positive knots. We prove that if a positive knot is alternating, then all it...
متن کامل