"Linking number"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==linking number and HOMFLY polynomial== | ==linking number and HOMFLY polynomial== | ||
| − | * Let | + | * Let <math>L</math> be a link. |
| − | * | + | * <math>P_L</math> denote the HOMFLY polynomial |
| − | * recall that | + | * recall that <math>P_L(a,z)\in \mathbb[a^{\pm 1}, z^{\pm 1}]</math> satisfies the skein relation |
:<math> | :<math> | ||
aP_{L_{+}} - a^{-1}P_{L_{-}}=zP_{L_0} | aP_{L_{+}} - a^{-1}P_{L_{-}}=zP_{L_0} | ||
</math> | </math> | ||
and | and | ||
| − | + | :<math> | |
P_{n-unlink}=\left(\frac{a-a^{-1}}{z}\right)^{n-1} | P_{n-unlink}=\left(\frac{a-a^{-1}}{z}\right)^{n-1} | ||
| − | + | </math> | |
;thm (Sikora) | ;thm (Sikora) | ||
| − | For any link | + | For any link <math>L</math> of <math>n</math> components the limit |
| − | + | :<math> | |
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})}) | 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})}) | ||
| − | + | </math> | |
exists. | exists. | ||
| − | + | <math>Q_L(q)</math> is a polynomial in <math>q</math> and <math>Q_L(q)=\sum c_i(L)q^i</math> | |
2020년 11월 16일 (월) 04:27 판
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