Knot and Link Invariants

Knot and Link Invariants

A knot (link) invariant is a function on the set of knots (links) that is constant on isotopy classes. That is, if and are equivalent up to ambient isotopy, then .

Remark

It is also true that implies , while different knots may have the same invariant.

Tricolourbility

A knot is tricolourable if there exists a proper 3-colouring of the knot diagram.

Jones Polynomial

Jones Polynomial

The Jones polynomial of a link is recursively defined as follows:

• for the unknot ;
• skein relation: , where , , and are three links, which are identical except for some neighbourhood, in which they differ by the crossing changes or smoothing according to the figure below:

Moreover, a modified Jones polynomial is obtained from the Jones polynomial by substituting .

Conway Polynomial

Alexander–Conway Polynomial

The Conway polynomial associates to each link a polynomial in . Given a link diagram , 𝐶(𝐷) obeys two rules:

• For the unknot diagram , ;