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An oriented knot is an oriented link of one component, or equivalently, it is a knot which has been given an orientation. Given an oriented knot K, reversing the orientation ...

A number of graphs are associated with P. J. Owens. The 76-node Owens graph (Owens 1980) provides the smallest known example of a polyhedral quintic nonhamiltonian graph. It ...

A relation "<=" is a partial order on a set S if it has: 1. Reflexivity: a<=a for all a in S. 2. Antisymmetry: a<=b and b<=a implies a=b. 3. Transitivity: a<=b and b<=c ...

Two knots are pass equivalent if there exists a sequence of pass moves taking one to the other. Every knot is either pass equivalent to the unknot or trefoil knot. These two ...

A number which is simultaneously a pentagonal number P_n and a square number S_m. Such numbers exist when 1/2n(3n-1)=m^2. (1) Completing the square gives ...

A number which is simultaneously a pentagonal number P_n and triangular number T_m. Such numbers exist when 1/2n(3n-1)=1/2m(m+1). (1) Completing the square gives ...

A test for the primality of Fermat numbers F_n=2^(2^n)+1, with n>=2 and k>=2. Then the two following conditions are equivalent: 1. F_n is prime and (k/F_n)=-1, where (n/k) is ...

A group that coincides with its commutator subgroup. If G is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that ...

The Perko pair is the pair of knots 10_(161) and 10_(162) illustrated above. For many years, they were listed as separate knots in Little (1885) and all similar tables, ...

A planted tree is a rooted tree whose root vertex has vertex degree 1. The number of planted trees of n nodes is T_(n-1), where T_(n-1) is the number of rooted trees of n-1 ...