Recently I have stumbled upon links which are closures of braids, of the form $\sigma = \tau^{n}$. Such links generalize torus links. Are there any papers studying such links? In particular I am interested in questions like which links appear in this way, what can we say about polynomial invariants of such links, are there any criteria in terms of polynomial links invariants.
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2 Answers
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If you only cared about knots, then you'd be looking at a special case of the so-called periodic knots, and the latter have been studied by several people.
There is this paper by Davis and Livingston where some results are summarised and some others are referred to in the introduction. In particular, they state an older result of Murasugi: the Alexander polynomial of the closure $K$ of $\sigma$ is determined by the (two-variable) Alexander polynomial of the link $L =\hat\tau\cup B$, where $B$ is the unknot that links the braid (the axis of the braid, as I believe it's called). Namely: $$ \delta_\ell(t)\cdot\Delta_K(t) = \prod_{j=1}^n\Delta_L(\zeta^j,t), $$ where $\ell$ is the number of strands of the braid, $\delta_\ell(t) = (1-t^\ell)/(1-t)$, $\zeta$ is a primitive $n$-th root of unity, and the first variable of $\Delta_L$ corresponds to the (meridian of) the unknotted component $B$.
answered Feb 10, 2015 at 16:29
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$\begingroup$ I am aware of this paper. However, I would like to distinguish between periodic knots and periodic knots, which are closures of some power of a braid. Anyways, thanks for the answer. $\endgroup$ Commented Mar 11, 2015 at 8:55
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This recent paper seems to answer your question partially. It shows that the limit of the Khovanov homology (and so in particular the Jones polynomial) behaves like that of the Jones-Wenzel projector as $n \to \infty$. This generalizes a result of Rozansky who proved the result for torus knots.
