On the signature of a positive braid
[Sur la signature d’une tresse positive]
Greene, Joshua Evan1 ; Liechti, Livio2
1Department of Mathematics, Boston College, Chestnut Hill, MA 02467, (United States of America)
2Département de Mathématiques, Université de Fribourg, Chemin du Musée 23, 1700 Fribourg (Switzerland)
Annales Henri Lebesgue, Tome 7 (2024), pp. 823-839

We show that the signature of a positive braid link is bounded from below by one-quarter of its first Betti number. This equates to one-half of the optimal bound conjectured by Feller, who previously provided a bound of one-eighth.

On montre que la signature d’un entrelacs représentable par une tresse positive est au moins un quart de son premier nombre de Betti. Cela correspond a la moitié de la borne optimale conjecturée par Feller, qui avait auparavant prouvé une borne d’un huitième.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/ahl.212
Classification : 57K10
Keywords: Signature, positive braid, chessboard surface, Goeritz form

Greene, Joshua Evan  1   ; Liechti, Livio  2

1 Department of Mathematics, Boston College, Chestnut Hill, MA 02467, (United States of America)
2 Département de Mathématiques, Université de Fribourg, Chemin du Musée 23, 1700 Fribourg (Switzerland)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AHL_2024__7__823_0,
     author = {Greene, Joshua Evan and Liechti, Livio},
     title = {On the signature of a positive braid},
     journal = {Annales Henri Lebesgue},
     pages = {823--839},
     year = {2024},
     publisher = {\'ENS Rennes},
     volume = {7},
     doi = {10.5802/ahl.212},
     mrnumber = {4799911},
     zbl = {07914807},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/ahl.212/}
}
TY  - JOUR
AU  - Greene, Joshua Evan
AU  - Liechti, Livio
TI  - On the signature of a positive braid
JO  - Annales Henri Lebesgue
PY  - 2024
SP  - 823
EP  - 839
VL  - 7
PB  - ÉNS Rennes
UR  - https://www.numdam.org/articles/10.5802/ahl.212/
DO  - 10.5802/ahl.212
LA  - en
ID  - AHL_2024__7__823_0
ER  - 
%0 Journal Article
%A Greene, Joshua Evan
%A Liechti, Livio
%T On the signature of a positive braid
%J Annales Henri Lebesgue
%D 2024
%P 823-839
%V 7
%I ÉNS Rennes
%U https://www.numdam.org/articles/10.5802/ahl.212/
%R 10.5802/ahl.212
%G en
%F AHL_2024__7__823_0
Greene, Joshua Evan; Liechti, Livio. On the signature of a positive braid. Annales Henri Lebesgue, Tome 7 (2024), pp. 823-839. doi: 10.5802/ahl.212

[BDL18] Baader, S.; Dehornoy, P.; Liechti, L. Signature and concordance of positive knots, Bull. Lond. Math. Soc., Volume 50 (2018) no. 1, pp. 166-173 | MR | DOI | Zbl

[Deh15] Dehornoy, P. On the zeroes of the Alexander polynomial of a Lorenz knot, Ann. Inst. Fourier, Volume 65 (2015) no. 2, pp. 509-548 | MR | DOI | Zbl | Numdam

[Fel15] Feller, P. The signature of positive braids is linearly bounded by their genus, Int. J. Math., Volume 26 (2015) no. 10, p. 1550081 | DOI | Zbl

[Fel18] Feller, P. A sharp signature bound for positive four-braids, Q. J. Math., Volume 69 (2018) no. 1, pp. 271-283 | MR | DOI | Zbl

[GL78] Gordon, C. McA.; Litherland, R. A. On the Signature of a Link, Invent. Math., Volume 47 (1978), pp. 53-69 | MR | DOI | Zbl

[GL16] Gilmer, P.; Livingston, C. Signature jumps and Alexander polynomials for links, Proc. Am. Math. Soc., Volume 144 (2016) no. 12, pp. 5407-5417 | MR | DOI | Zbl

[Goe33] Goeritz, L. Knoten und quadratische Formen, Math. Z., Volume 36 (1933), pp. 647-654 | MR | DOI | Zbl

[KT76] Kauffman, L. H.; Taylor, L. R. Signature of links, Trans. Am. Math. Soc., Volume 216 (1976), pp. 351-365 | MR | DOI | Zbl

[Lic97] Lickorish, W. B. R. An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer, 1997 | MR | Zbl | DOI

[Lie16] Liechti, L. Signature, positive Hopf plumbing and the Coxeter transformation (With appendix by Peter Feller and Livio Liechti), Osaka J. Math., Volume 53 (2016) no. 1, pp. 251-266 | MR | Zbl

[Lie20] Liechti, L. On the genus defect of positive braid knots, Algebr. Geom. Topol., Volume 20 (2020) no. 1, pp. 403-428 | MR | DOI | Zbl

[Mur65] Murasugi, K. On a certain numerical invariant of link types, Trans. Am. Math. Soc., Volume 117 (1965), pp. 387-422 | MR | DOI | Zbl

[Rud82] Rudolph, L. Non-trivial positive braids have positive signature, Topology, Volume 21 (1982), pp. 325-327 | MR | Zbl | DOI

[Sta78] Stallings, J. Constructions of fibred knots and links, Algebraic and Geometric Topology (Proceedings of Symposia in Pure Mathematics), Volume 32, American Mathematical Society (1978), pp. 55-60 | Zbl | DOI

[Sto08] Stoimenow, A. Bennequin’s inequality and the positivity of the signature, Trans. Am. Math. Soc., Volume 360 (2008) no. 10, pp. 5173-5199 | MR | DOI | Zbl

[Tro62] Trotter, H. F. Homology of group systems with applications to knot theory, Ann. Math., Volume 76 (1962), pp. 464-498 | DOI | Zbl

Cité par Sources :