The second coefficient of the Alexander polynomial as a satellite obstruction

Lukas Lewark

doi:10.5802/crmath.694

Research article - Geometry and Topology

The second coefficient of the Alexander polynomial as a satellite obstruction

Lukas Lewark ¹

¹ ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 7-11.

Abstract (OV)
Résumé

A set $\mathcal{P}$ of links is introduced, containing positive braid links as well as arborescent positive Hopf plumbings. It is shown that for links in $\mathcal{P}$, the leading and the second coefficient of the Alexander polynomial have opposite sign. It follows that certain satellite links, such as $(n,1)$-cables, are not in $\mathcal{P}$.

Un ensemble $\mathcal{P}$ de liens est introduit, contenant les clôtures de tresses positives ainsi que les plombages arborescents de bandes de Hopf positives. Il est démontré que pour les liens appartenant à $\mathcal{P}$, le premier et le deuxième coefficient du polynôme d’Alexander sont de signe opposé. Il s’ensuit que certains liens satellites, tels que les câbles $(n,1)$, n’appartiennent pas à $\mathcal{P}$.

Received: 2024-03-11
Revised: 2024-10-09
Accepted: 2024-10-13
Published online: 2025-03-06

DOI: 10.5802/crmath.694

Classification: 57K10, 57K14
Keywords: Alexander polynomial, Hopf plumbings, satellite knots, arborescent knots, positive braids
Mots-clés : Polynôme d’Alexander, plombages de Hopf, nœuds satellites, nœuds arborescents, tresses positives

Author's affiliations:

Lukas Lewark ¹

¹ ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {The second coefficient of the {Alexander} polynomial as a satellite obstruction},
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     language = {en},
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Lukas Lewark. The second coefficient of the Alexander polynomial as a satellite obstruction. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 7-11. doi : 10.5802/crmath.694. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.694/

References
Cited by

[1] Emil Artin Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb., Volume 4 (1925) no. 1, pp. 47-72 | DOI | MR | Zbl

[2] Sebastian Baader Hopf plumbing and minimal diagrams, Comment. Math. Helv., Volume 80 (2005) no. 3, pp. 631-642 | DOI | MR | Zbl

[3] Francis Bonahon; Laurence C. Siebenmann New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots (2016) https://dornsife.usc.edu/francis-bonahon/preprints-available/

[4] Allan L. Edmonds; Charles Livingston Group actions on fibered three-manifolds, Comment. Math. Helv., Volume 58 (1983) no. 4, pp. 529-542 | DOI | MR | Zbl

[5] David Gabai Genera of the arborescent links, Mem. Am. Math. Soc., Volume 339 (1986), pp. 1-98 | MR | Zbl

[6] Tetsuya Ito A note on HOMFLY polynomial of positive braid links, Int. J. Math., Volume 33 (2022) no. 4, 2250031, 18 pages | DOI | MR | Zbl

[7] Tetsuya Ito Satellite fully positive braid links are braided satellite of fully positive braid links (2024) | arXiv

[8] Siddhi Krishna Taut foliations, braid positivity, and unknot detection (2023) | arXiv

[9] W. B. Raymond Lickorish An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer, 1997, x+201 pages | DOI | MR | Zbl

[10] Lee Rudolph Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv., Volume 58 (1983), pp. 1-37 | DOI | MR | Zbl

[11] Lee Rudolph Quasipositive plumbing (constructions of quasipositive knots and links. V), Proc. Am. Math. Soc., Volume 126 (1998) no. 1, pp. 257-267 | DOI | MR | Zbl

[12] Lee Rudolph Knot theory of complex plane curves, Handbook of knot theory, Elsevier, 2005, pp. 349-427 | DOI | MR | Zbl

[13] John R. Stallings Constructions of fibred knots and links, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2 (Proceedings of Symposia in Pure Mathematics), Volume XXXII, American Mathematical Society, 1978, pp. 55-60 | MR | Zbl

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