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
expositions
메타데이터
위키데이터
- ID : Q2000614