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Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.

Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?

Here, $J_L(q)$ is the jones polynomial of a link $L$ and $\hat{\sigma}$ stands for the closure of a braid $\sigma$.

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Maybe there are, in some special cases. But, for example, if you take $\sigma_1 = \sigma_2$ to be 1/n-th of a twist, then $\widehat{\sigma}_1 = \widehat{\sigma}_2 = T_{1,n}$ are both unknotted, but the closure of their concatenation is $T_{2,n}$, which has nontrivial Jones polynomial. –  Marco Golla May 19 at 9:49
    
Even better, if you take $\sigma_1 = \sigma_2^{-1}$, then you need to find a relationship between the trivial polynomial, and $V(q), V(q^{-1})$ for some Jones polynomial $V$. –  Marco Golla May 19 at 9:55
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It may be worth mentioning that there is a map on Khovanov homology relating the closures of ${\sigma}_1, {\sigma}_2$, and ${\sigma}_1{\sigma}_2$. The map corresponds to the cobordism which consists of $n$ saddles joining the respective strands of ${\sigma}_1, {\sigma}_2$. This doesn't give a formula relating the Jones polynomial (=graded Euler characteristic) of the three links, but this does provide some explanation for the various special cases mentioned in a few other comments and answers. –  Slava Krushkal May 19 at 20:13

2 Answers 2

up vote 13 down vote accepted

Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial $J_\sigma$ (roughly) comes from taking the trace of a linear map $A_\sigma$ associated to the braid $\sigma$, so the question (roughly) asks about relations between $Tr(A_\sigma)$, $Tr(A_\tau)$, and $Tr(A_\sigma A_\tau)$.

Let $a,b$ generate a free group $\pi$, and let $Chr$ be the $SL_2$ character scheme of $\pi$ (i.e. the space of pairs of $SL_2$ matrices considered up to simultaneous conjugation). Define $Tr(g):Chr \to \mathbb C$ to be the function $\rho \mapsto Tr(\rho(g))$. Then the functions $Tr(a)$, $Tr(b)$, and $Tr(ab)$ are algebraically independent. So there shouldn't be any universal formula relating these three traces, whether $a,b \in SL_2$ or are bigger matrices.

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In the very special case where $\sigma_2$ is a full twist of all $n$ strands, there is some information. This is essentially because full twists are in the center of the braid group. See this paper by Champanerkar and Kofman for details:

http://www.math.csi.cuny.edu/~ikofman/jp_gap_PAMS.pdf

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