关于Borromean Link的Jones Polynomial正确性及VL(1)计算结果的疑问

Hi there! Great question—let's break this down clearly since you're an enthusiast diving into knot theory.

First off: The polynomial you found from the knot atlas is indeed the correct Jones polynomial for the Borromean link. So that part checks out completely!

Now, to address your confusion about ( V_L(1) = 4 ) instead of 3 (the number of components): Your assumption that evaluating the Jones polynomial at ( q=1 ) gives the number of components is a common mix-up, but it only applies to the unlink—a set of completely separate, unconnected knots/links. For non-split links like the Borromean rings (where components are interdependent; you can't separate any two without breaking the third), this property doesn't hold.

Let me elaborate a bit more:

• For the unlink of ( n ) components, the Jones polynomial does evaluate to ( (-1)^{n-1} \times n ) at ( q=1 ). So for a 3-component unlink, that would be ( 1 \times 3 = 3 ), which matches your initial expectation.
• But the Borromean link isn't an unlink. It's a unique "fully linked" structure where no two components have a direct linking number (they don't loop around each other on their own), but all three rely on each other to stay connected. This intricate interdependency is encoded in the Jones polynomial, which is why ( V_L(1) = 4 ) instead of 3.

Put simply: The Jones polynomial isn't just a count of components—it captures deeper topological information about how the link is woven together. The Borromean link's polynomial reflects this non-trivial linking structure, so its value at ( q=1 ) can't be reduced to just the number of parts.

If you want to confirm the number of components for a link, you'd typically rely on simpler topological checks (like visual counting, or invariants explicitly designed to track component count) rather than the Jones polynomial at ( q=1 ).

备注：内容来源于stack exchange，提问作者ParabolicX