The recursive defining relation of the Conway polynomial is a special instance of the following more general relation with two parameters, $v$ and $z$.

where ${L}_{+},{L}_{-},{L}_{0}$ are interpreted as before for a given link $L$ with a distinguished crossing. This relation is an example of a skein relation .

Again, setting $H\left(L\right)=1$ if L is the trivial oriented knot, we can compute the value of $H$ on any oriented link by recursion and the value on trivial links, which are computed in an exercise.

The result is called the HOMFLY polynomial . The relation can be applied to any crossing in a link diagram, so the application of the skein relation is not deterministic. In particular, it is not immediate that the relations uniquely determine a polynomial.

The skein relation can be interpreted as an element of the group algebra $\mathbb{Z}[v,v^{-1},z][B_n]$ by viewing the oriented link as the closure of a braid $b$ on $n$ strands. Write $b$ as a product of elementary braids. The crossing under scrutiny in the skein relation then occurs as an element $s_i$ for some $i\in[n-1]$. The relation reads $v^{-1}s_i - v s_i^{-1} = z$.

For the study of the invariant, we need only work in the quotient ring $A_n$ of $\mathbb{Z}[v,v^{-1},z][B_n]$ by the ideal generated by all elements of the form $s_i^2 - zv s_i - v^2$.

We will write $t_i$ for the image in $A_n$ of $s_i$. The assignment $s_i\mapsto t_i$ determines a group homomorphism $B_n\to (A_n)^{\times}$, where, for a ring $R$, we write $R^{\times}$ for its group of invertible elements.

The algebra $A_n$ is a free $\mathbb{Z}[v,v^{-1},z]$-module of rank $n!$. Let $T_n$ be a collection of minimal expressions in the generators $t_i$ for words corresponding to permutations, one for each permutation. Then $T_n$ is a basis of $A_n$. In particular each element of $T_n$ can be written as either an element of $T_{n-1}$ or is of the form $at_{n-1}b$ with $a,b\in T_{n-1}$.

In view of this isomorphism, the homomorphism of rings $A_n\to A_{n+1}$ determined by sending $t_i$ in $A_n$ to the generator with the same name in $t_i$ in $A_{n+1}$ is injective.

We add a variable $t$ to our coefficient ring and require the relation between our variables

Write $S$ for this ring, that is, the quotient ring of $\mathbb{Z}[v,v^{-1},z,t, t^{-1} ]$ by the ideal generated by $v^{-1}t - v t^{-1} - z$.

We claim that there exist linear maps $\tau_n : A_{n}\to S$ such that

For $n = 0$ we interpret $A_0$ as $S$. By b. and c., $\tau_0$ must be the identity on $A_0$.

We proceed by induction, assuming that the statements holds for $\tau_k$ with $k\le n$. In view of linearity, its suffices to describe $\tau_{n+1}$ on the basis elements of the form $at_nb$ with $a,b\in T_{n}$. Here we should have $\tau_{n+1}(at_nb) = t \tau_n(ab)$ and so we take this as our definition.

It remains to show $\tau_{n+1}(xy) = \tau_{n+1}(yx)$ for $x,y\in T_{n+1}$. We deal with one example where $x = a t_{n} b$ and $y = t_n$, and $b$ are not in $T_{n-1}$. Then there are $u,v,p,q\in T_{n-1}$ such that $a = ut_{n-1}v$ and $b = pt_{n-1}q$. Now $\tau_{n+1}(at_nbt_n) = \tau_{n+1}(at_npt_{n-1}qt_n) = \tau_{n+1}(apt_nt_{n-1}t_nq)$ $= \tau_{n+1}(apt_{n-1}t_nt_{n-1}q) = t\tau_{n}(apt_{n-1}^2q) = zt\tau_{n}(apt_{n-1}q) + vt\tau_{n}(apq)$ $=zt\tau_{n}(ab) + vt\tau_{n}(ut_{n-1}vpq) = zt\tau_{n}(ab) + vt^2\tau_{n}(uvpq)$.

Similarly, $\tau_{n+1}(t_nat_nb) = zt\tau_{n}(ab) + vt^2\tau_{n}(uvpq)$. So, indeed, $\tau_{n}(at_nbt_n) = \tau_{n}(t_nat_nb)$.

For $b\in B_n$, write $\tau(b) = t^{-h(b)}\tau_n(b)$, where $h$ is the height. The scalar helps us to obtain invariance under the second Markov move. Indeed, $\tau(bs_n) = t^{-h(b)-1} \tau_{n+1}(bt_n) = t^{-h(b)} \tau_{n}(b) = \tau_n(b)$ and $\tau(bs_n^{-1}) = t^{-h(b)-1} \tau_{n+1}(bt_n^{-1}) = t^{-h(b)-1} \tau_{n+1}(v^{-2}bt_n-zv^{-1}b)$ $= v^{-2}t^{-h(b)-1} \tau_{n+1}(bt_n) -zv^{-1} t^{-h(b)-1} \tau_{n}(b)$ $= (v^{-2}t-zv^{-1})t\tau(b) =1$.

Now $\tau$ satisfies $\tau(as_i^{-1}b)= t^{-h(ab)+1}\tau_n(at_i^{-1}b)= v^{-2}t^{-h(ab)+1}\tau_n(at_ib) -v^{-1}zt^{-h(ab)+1}\tau_n(ab)$ $v^{-2}t^2\tau(as_ib) -v^{-1}zt\tau(ab)$. This rewrites to

for braids $c_{+}, c_{-}, c_0$ whose closure correspond to links $L_{+}, L_{-}, L_0$ mutually related as before. Substituting $v$ by $tv$ and $z$ by$vz$, we find the HOMFLY recursion.

If we specialize $v$ to $1$ and $z$ to ${t}^{\frac{-1}{2}}-{t}^{\frac{1}{2}}$ in the HOMFLY polynomial, we obtain the Conway polynomial.

If we specialize $v$ to $t$ and $z$ to ${t}^{\frac{1}{2}}-{t}^{\frac{-1}{2}}$ in the HOMFLY polynomial, we obtain the Jones polynomial .

The Kauffman bracket gives another way of computing the Jones polynomial of a knot as a sum of states obtained from splitting each crossing in two different ways.

$\langle K \rangle = \sum_SA^{a(S)}B^{b(S)}d^{c(S)-1}$, where the sum is over the states of $K$ and $a(S)$ is the number of $A$ splittings in $S$, $b(S)$ is the number of $B$ splittings in $S$, and $c(S)$ is the number of components in $S$.

If $AB = 1$ and $A^2+B^2+ABd = 0$, then $f(K)(A) = (-A)^{-3w(K)}\langle K \rangle$, where $w(K)$ is the writhe of $K$, is invariant under Reidemeister II and III.

HOMEWORK CHALLENGE 1 (3 pts): prove this theorem.

The fundamental group of a topological space is a well known concept in algebraic topology.

For a knot, the topological space of greatest interest is the complement of the knot in the Euclidean space $E$. It is a knot invariant. A presentation of the group in terms of generators and relations can be given.

The Wirtinger presentation of an oriented link diagram $D$ is the following presentation of the fundamental group of the link $L$ corresponding to $D$. Its generators are the arcs of $D$. Suppose there are $n$ arcs. Let ${g}_{i}$ for $i\in \left\{1,\mathrm{...},n\right\}$ denote the arcs.

For each oriented crossing with overpassing arc ${g}_{k}$, underpassing entering arc ${g}_{i}$ and underpassing emerging arc ${g}_{j}$, we add the relation