Linking number

Pythagoras0 (토론 | 기여)님의 2020년 12월 28일 (월) 00:18 판 (→‎메타데이터: 새 문단)
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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

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