Towards tangle calculus for Khovanov polynomials

Abstract

We provide new evidence that the tangle calculus and "evolution" are applicable to the Khovanov polynomials for families of long braids inside the knot diagram. We show that jumps in evolution, peculiar for superpolynomials, are much less abundant than it was originally expected. Namely, for torus and twist satellites of a fixed companion knot, the main (most complicated) contribution does not jump, all jumps are concentrated in the torus and twist part correspondingly, where these jumps are necessary to make the Khovanov polynomial positive. Among other things, this opens a way to define a jump-free part of the colored Khovanov polynomials, which differs from the naive colored polynomial just "infinitesimally". The separation between jumping and smooth parts involves a combination of Rasmussen index and a new knot invariant, which we call "Thickness".

Figures (3)

• Figure 1: Two 2-cabled knot diagrams $\tilde{\cal K}$, cut at arbitrary place, with the free ends glued together in two different ways to form a new knot. Position of cut and new $n$ crossings does not matter (all choices are topologically/Reidemeister equivalent). Notations are adjusted to allow a unified notation ${\cal S}_{T_n}^{\tilde{\cal K}}$ for both cases. Note that we fix the parity of $n$ crossings to be odd in the case of torus satellites and even in the case of twist satellites to choose formulas for $\mu$ and $\tau$ elements, see subsequent sections. Arrows show orientation of a resulting knot. Twisting tangle is denoted by $\cal T$. A possible content of the box is shown in Fig. \ref{['fig:2']}. Note that this definition involves not just the knot ${\cal K}$, but a particular knot diagram $\tilde{\cal K}$. However, dependence on the diagram is simple: it can be eliminated by s shift of $n$ by $2w_{\tilde{\cal K}}$, where $w_{\tilde{\cal K}}$ is the writhe number of the diagram (the difference between the numbers of positive and negative crossings in $\tilde{\cal K}$): ${\cal S}_{T_n}({\cal K}):={\cal S}_{T_{n-2w_{\tilde{\cal K}}}}^{\tilde{\cal K}}$ does not depend on the diagram.
• Figure 2: 2-cabled trefoil as an example of possible content of the box in Fig. \ref{['fig:1']}. See Fig. \ref{['fig:3']} below for possible 2-strand and 3-strand realizations of this trefoil, which are convenient for calculations in the case of HOMFLY polynomials and clarify the structure of the answer in the Khovanov case.
• Figure 3: Two realizations of the trefoil from Fig. \ref{['fig:2']}. The trefoil itself can be represented as a 2-strand braid with each strand in representation $[2]\oplus [1,1]$ or $\varnothing\oplus {\rm adj}$. The corresponding knot polynomial is then decomposed in contributions of irreducible representations from $[2]\otimes[2]$ and $[1,1]\otimes[1,1]$ or $\varnothing\otimes\varnothing=\varnothing$ and a triple from ${\rm adj}\otimes {\rm adj}$. Alternative representation of the same trefoil is 3-strand. Then cubes of representations appear instead of squares. If the trefoil is substituted by an arbitrary $m$-strand knot, representations contributing to a knot polynomial are irreducible representations from the $m$-th powers $[2]^{\otimes m}$, $[1,1]^{\otimes m}$, $\varnothing^{\otimes m}=\varnothing$, ${\rm adj}^{\otimes m}$. In the case of Jones and Khovanov polynomials ($N=2$) for satellite knots (not links) the only non-trivial contribution is from ${\rm adj}^{\otimes m}=[2]^{\otimes m}$.