# "Linking number"의 두 판 사이의 차이

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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