FAST KHOVANOV HOMOLOGY COMPUTATIONS

By Dror Bar-Natan. 2006

193 citations recorded by the source corpus.

We introduce a local algorithm for Khovanov homology computations — that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational efficiency. Thus our program can rapidly compute certain Khovanov homology groups that otherwise would have taken centuries to evaluate.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b6766dd60cdc4bf9da68916729eb1dcccdd9583c

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A spectral sequence for Khovanov homology with an application to (3,q)–torus links (2006, methodology): “That said, there are some ingenious algorithms and methods which now allow computer calculations in good cases for diagrams with 60 or so crossings (see BarNatan [3]).”
Fixing the functoriality of Khovanov homology (2007, methodology): “We’ll construct the chain maps for the first R3 move directly using the simplification algorithm described by Bar-Natan in [3]; specifically, applying it to the complexes”
Man and machine thinking about the smooth 4-dimensional Poincaré conjecture (2009, methodology): “We started with a program written in Java by Green [21], implementing Bar-Natan’s algorithm described in [7].”
Torsion of Khovanov homology (2014, unknown): “The ﬁrst known counter-example is the (5 , 6)-torus knot [BN2] with 3-torsion.”
Braiding via geometric Lie algebra actions (2010, methodology): “hism on k summands of the form A[·]” if the sum (over all n) of the A[n]-ranks of f is k. 4.1.3. Gaussian elimination. Finally, we will repeatedly use the following cancellation Lemma which Bar-Natan [BN] calls “Gaussian elimination”. Lemma 4.1. Let X,Y,Z,W be four objects in a triangulated category”
Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case (2007, unknown): “=Ui[−1]{2}. Hence Ui∗Ui∼=Pi β∗P i βR∗P i β∗P i βR [2]{−4}. From the proof of Lemma 5.9, we have that Pi βR ∗Pi∼=O ∆ ⊕O ∆[−2]{2}and the result follows. We also need the following lemma which Bar-Natan [BN] calls “Gaussian elimination”. Lemma 5.25. Let X,Y,Z,Wbe four objects in a triangulated category”