• Corpus ID: 207796449

Immersed curves in Khovanov homology

@article{Kotelskiy2019ImmersedCI,
  title={Immersed curves in Khovanov homology},
  author={Artem Kotelskiy and Liam Watson and Claudius Zibrowius},
  journal={arXiv: Geometric Topology},
  year={2019}
}
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of… 
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References

SHOWING 1-10 OF 105 REFERENCES
Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
We suggest a categorification procedure for the SO(2N) one-variable specialization of the two-variable Kauffman polynomial. The construction has many similarities with the HOMFLYPT categorification:
Khovanov's homology for tangles and cobordisms
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological
Peculiar modules for 4‐ended tangles
With a 4‐ended tangle T , we associate a Heegaard Floer invariant CFT∂(T) , the peculiar module of T . Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University,
A refinement of Rasmussen's s-invariant
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these
Geom
  • Topol., 2:665–741,
  • 2002
A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifolds
We extend the definition of Khovanov-Lee homology to links in connected sums of $S^1 \times S^2$'s, and construct a Rasmussen-type invariant for null-homologous links in these manifolds. For certain
Proceedings of Gökova Geometry-Topology Conference 2002
2019
On symmetries of peculiar modules, or $\delta$-graded link Floer homology is mutation invariant
We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete
MATH
TLDR
It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/z, and some other physical models of knots as flattened ropes or strips which exhibit similar length versus complexity power laws are surveyed.
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