Question

knot theory

(a). what is a link? give some examples

(b) With examples explain invariants of links

Answer 1

Solution : (a) Link : Link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component.

Example : For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles.

(b)  Invariants of link : Invariant of link is a function from the set of all links to any other set such that the function does not change as the link is changed . In other words, invariant of link always assigns the same value to equivalent links (although different knots may have the same link invariant).

Examples of link invariant : Let , c( L ) = no. of components of link L .

Then c( Un) = n , but c( K) = 1 for every knot K

Now,  k( L ) = no. of ‘cuts’ needed to ‘unlink’ L, but  saying k ( K) = 0 is no easier than saying K ∼ U1!.

Therefore , useful link invariants must be discriminating and computable.