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In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope [Formula: see text] such that [Formula: see text]-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.
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Cyclic branched covers of Seifert links and properties related to the ADE$ADE$ link conjecture
Abstract In this article, we show that all cyclic branched covers of a Seifert link have left‐orderable fundamental groups, and therefore admit co‐oriented taut foliations and are not ‐spaces, if and only if it is not an link up to orientation. This leads to a proof of the link conjecture for Seifert links. When is an link up to orientation, we determine which of its canonical ‐fold cyclic branched covers have nonleft‐orderable fundamental groups. In addition, we give a topological proof of Ishikawa's classification of strongly quasi‐positive Seifert links and we determine the Seifert links that are definite, resp., have genus zero, resp. have genus equal to its smooth 4‐ball genus, among others. In the last section, we provide a comprehensive survey of the current knowledge and results concerning the link conjecture.
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- Award ID(s):
- 2409398
- PAR ID:
- 10610843
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of the London Mathematical Society
- Volume:
- 111
- Issue:
- 6
- ISSN:
- 0024-6107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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