Odd-homology thickness conjecture for mixed rational-tangle algebraic links

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Let pip_i and qjq_j be positive rational tangles not starting with 11, and consider (p1,…,pm)(q1,…,qn−)(p_1,\ldots,p_m)(q_1,\ldots,q_n-). Assume the tangles are not all integer. Let p‾i\overline p_i and q‾j\overline q_j be their reductions, and put q=min⁡(q‾1,…,q‾n)q=\min(\overline q_1,\ldots,\overline q_n).

Mixed-tangle thickness conjecture. The link is Kh‾′\overline{Kh}'-thick if m≥3m\ge3, q>1q>1, each pip_i of length at least 22 satisfies p‾i=1\overline p_i=1, and

max⁡(p‾1,p‾2,…,p‾m)≤q.\max(\overline p_1,\overline p_2,\ldots,\overline p_m)\le q.

This extends the preceding integer-tangle criterion to mixed rational tangles. The supplied text gives no resolution status.

References

Primary source

Slavik Jablan and Radmila Sazdanović, “Quasi-alternating links and odd homology: computations and conjectures”, arXiv:0901.0075 (2014).

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