Metadata
Abstract
We solve an asymptotic variant of a smooth version of the $A_n$-realization problem for plane curves. As an application, we determine the cobordism distance between torus links of type $T(d,d)$ and $T(2,N)$ up to an error of at most $3d$. We also discuss the limits of knot theoretic approaches aimed at solving the $A_n$-realization problem.
References
[Baa12] Scissor equivalence for torus links, Bull. Lond. Math. Soc., Volume 44 (2012) no. 5, pp. 1068-1078 | MR | DOI | Zbl
[BCG17] A note on cobordisms of algebraic knots, Algebr. Geom. Topol., Volume 17 (2017) no. 4, pp. 2543-2564 | DOI | MR | Zbl
[BFLZ19] Khovanov width and dealternation number of positive braid links, Math. Res. Lett., Volume 26 (2019) no. 3, pp. 627-641 | DOI | MR | Zbl
[Fel16] Optimal cobordisms between torus knots, Commun. Anal. Geom., Volume 24 (2016) no. 5, pp. 993-1025 | DOI | MR | Zbl
[FP21] Genus one cobordisms between torus knots, Int. Math. Res. Not., Volume 2021 (2021) no. 1, pp. 523-550 | DOI | MR | Zbl
[GLM81] Signatures of covering links, Can. J. Math., Volume 33 (1981) no. 2, pp. 381-394 | DOI | MR | Zbl
[GLS98] Plane curves of minimal degree with prescribed singularities, Invent. Math., Volume 133 (1998) no. 3, pp. 539-580 | DOI | MR | Zbl
[GS20] Plane algebraic curves with prescribed singularities (2020) | arXiv
[Hir92] Construction of plane curves with cusps, Saitama Math. J., Volume 10 (1992), pp. 21-24 | MR | Zbl
[HW16] Four-ball genus bounds and a refinement of the Ozváth–Szabó tau invariant, J. Symplectic Geom., Volume 14 (2016) no. 1, pp. 305-323 | DOI | MR | Zbl
[KM93] Gauge theory for embedded surfaces. I, Topology, Volume 32 (1993) no. 4, pp. 773-826 | DOI | MR | Zbl
[Lew14] Rasmussen’s spectral sequences and the -concordance invariants, Adv. Math., Volume 260 (2014), pp. 59-83 | DOI | MR | Zbl
[Liv04] Computations of the Ozváth–Szabó knot concordance invariant, Geom. Topol., Volume 8 (2004), pp. 735-742 | DOI | MR | Zbl
[Mur65] On a certain numerical invariant of link types, Trans. Am. Math. Soc., Volume 117 (1965), pp. 387-422 | DOI | MR | Zbl
[Ore12] Some examples of real algebraic and real pseudoholomorphic curves, Perspectives in analysis, geometry, and topology (Progress in Mathematics), Volume 296, Birkhäuser, 2012, pp. 355-387 | DOI | MR | Zbl
[OS03] Knot Floer homology and the four-ball genus, Geom. Topol., Volume 7 (2003), pp. 615-639 | DOI | MR | Zbl
[OSS17] Concordance homomorphisms from knot Floer homology, Adv. Math., Volume 315 (2017), pp. 366-426 | DOI | MR | Zbl
[Ras10] Khovanov homology and the slice genus, Invent. Math., Volume 182 (2010) no. 2, pp. 419-447 | DOI | MR | Zbl
[Tro62] Homology of group systems with applications to knot theory, Ann. Math. (2), Volume 76 (1962), pp. 464-498 | DOI | MR | Zbl
[Wal04] Singular points of plane curves, London Mathematical Society Student Texts, 63, Cambridge University Press, 2004 | DOI | MR | Zbl