Linking number

linking number and HOMFLY polynomial

• Let $L$ be a link.
• $P_L$ denote the HOMFLY polynomial
• recall that $P_L(a,z)\in \mathbb[a^{\pm 1}, z^{\pm 1}]$ satisfies the skein relation

\[ aP_{L_{+}} - a^{-1}P_{L_{-}}=zP_{L_0} \] and $$ P_{n-unlink}=\left(\frac{a-a^{-1}}{z}\right)^{n-1} $$

thm (Sikora)

For any link $L$ of $n$ components the limit $$ Q_L(q) : = \lim_{v\to 1} \left(\frac{q}{a-a^{-1}}\right)^{\frac{n-1}{2}}P_L(a,\sqrt{q(a-a^{-1})}) $$ exists.

$Q_L(q)$ is a polynomial in $q$ and $Q_L(q)=\sum c_i(L)q^i$

• Birman
  • two 3-braids whose closures have the same Homfly-pt polynomial but different linking numbers between their components
  • pair of links with the same HOMFLYPT polynomial but different linking matrix

related items

• HOMFLY polynomial

expositions

• Sikora, Note on the Homfly-pt polynomial and linking numbers