Crosscap Numbers of Two-component Links

Crosscap Numbers of Two-component Links

We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, nonorientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.

Stick Numbers of Links with Two Trivial Components

A knot is an embedding of S in S, and a link is an embedding of one or more copies of S in S. The number of copies of S is called the number of components of the link. We usually think of a link as made out of string, but we can also think of the link as made up of line segments, which we call sticks, which can connect at any angle, but cannot bend. We would like to know the minimum number of s...

The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image Σ(L) of the natural homomorphism from the standard symmetry group MCG(S, L) to the product MCG(S) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the gr...

Insufficiency of Torres' Conditions for Two-component Classical Links

Torres has given necessary conditions for a polynomial to be the Alexander polynomial of a two component link. For certain links, additional conditions are necessary. Hillman gave one example for linking number 6. Here we give examples for all other linking numbers except 0, + 1, and +2.

Addition two generalized fuzzy numbers