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

메타데이터

위키데이터

• ID : Q2000614

Spacy 패턴 목록

• [{'LOWER': 'linking'}, {'LEMMA': 'number'}]