Knot and Link Invariants

Knot and Link Invariants

A knot (link) invariant is a function $f$ on the set of knots (links) that is constant on isotopy classes. That is, if $K_1$ and $K_2$ are equivalent up to ambient isotopy, then $f(K_1)=f(K_2)$.

Remark

It is also true that $f(K_1)\ne f(K_2)$ implies $K_1\ne K_2$, while different knots may have the same invariant.

Tricolourbility

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

Jones Polynomial

Jones Polynomial

The Jones polynomial of a link $J(L)(t)$ is recursively defined as follows:

• $J(O)=1$ for the unknot $O$;
• skein relation: $t^{-1}J(L_{+})-tJ(L_{-})=\left(t^{\frac{1}{2}}-t^{-\frac{1}{2}}\right)J(L_0)$, where $L_{+}$, $L_{-}$, and $L_0$ 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 $J(L)(h)$ is obtained from the Jones polynomial by substituting $t=e^h$.

Conway Polynomial

Alexander–Conway Polynomial

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

• For the unknot diagram $O$, $C(O)=1$;
• $C()-C()=z\cdot C()$