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If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form

enter image description here

(knots that are the same except inside a small disk, where they look like the pictures indicated).

In every case the result had a factor $(l^4 + 2l^2+1-l^2m^2)$.

My question is: why does the expression always have a factor $(l^4 + 2l^2+1-l^2m^2)$?

I understand that this happens when one of the links are disjoint due to the HOMFLY relation relating the disjoint sum (split union) and the connected sum: $$P(L_1 \sqcup L_2)=-\frac{l+l^{-1}}{m} P(L_1 \# L_2),$$ since if you stick this in, you get exactly the factor $(l^4 + 2l^2+1-l^2m^2)$.

Does the relation enter image description here perhaps hold in general? According to the proof for the connected sum formula, it shouldn't.

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    $\begingroup$ I haven't thought through the details, but HOMFLY describes quantum "GL_t" where t is allowed to vary continuously. When t happens to be an integer this has a quotient to the category of representations of quantum GL_n. The phenomenon you're observing here seems to be related to the quotient to GL_1. That is, for GL_1 the defining representation is invertible and so you expect to get the kind of relation you're looking at. $\endgroup$ Commented May 28, 2020 at 21:51

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First, doing a Reidemeister II move in the first diagram in your expression and then subtracting a multiple of the skein relation converts it to the change in the HOMFLY polynomial through a crossing change $P(L_+) - P(L_-)$, so your question is equivalent to asking why this difference is always divisible by $l^4 + 2l^2 + 1 - l^2 m^2$. Alternatively, since any two links with the same number of components differ by some sequence of crossing changes, we want $$P(L) \cong P(L') \mod (l^4 + 2l^2 + 1 - l^2m^2)$$ whenever $L$ and $L'$ have the same number of components.

Now, there are two values of the variable $m$ at which the HOMFLY polynomial is much simpler. If we substitute $m = l + l^{-1}$, then the quantity $(-1)^{c(L) - 1}$ satisfies the skein relation ($c(L)$ is the number of components), and if we substiture $m = -l-l^{-1}$ then the quantity $1$ satisfies the skein relation. Therefore, the residue of $P(L)$ modulo either $m-l-l^{-1}$ or $m+l+l^{-1}$ depends only on $c(L)$, so the same is true modulo $$\operatorname{gcd}(m-l-l^{-1},m+l+l^{-1}) = m^2 - (l+l^{-1})^2$$

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  • $\begingroup$ At the decategorified level all these spectral sequences should go away and you should just have honest quotients after specializing the variables. $\endgroup$ Commented May 28, 2020 at 21:51
  • $\begingroup$ Which conjecture(s) are you refering to? 1.5? Does my statement hold or is it still a conjecture? $\endgroup$
    – Jake B.
    Commented May 31, 2020 at 13:25
  • $\begingroup$ It's mostly conjecture 3.1 that's needed for the argument as written, but Noah Snyder's comment correctly points out that there's a much simpler unconditional argument. I'll edit shortly with more details. $\endgroup$ Commented May 31, 2020 at 20:42

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