Let $T\_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T\_{2,3} \sharp T\_{3,4}$ of the (2,3) and (3,4) torus knots? The (3,4) torus knots $T\_{3,4}$ is homologically thick. Is the Khovanov $\mathfrak{sl}(2)$ homology of $T\_{2,3} \sharp T\_{3,4}$ homologically thick?



Khovanov $sl_2$ homology of a connected sum of some torus knots
KnotTheory` gives the following Khovanov polynomial for the knot $K = T\_{2,3}\# T\_{3,4}$ you're looking at:
$$q^7 + q^9 + 2q^{11}t^2 + 2q^{15}t^3 + (q^{13} + 2q^{15})t^4 + (2q^{17} + 2q^{19})t^5 + (q^{17} + q^{21})t^6 + 2q^{21}t^7 + q^{25}t^8$$
The two summands have $s$ equal 2 and 6, therefore $s(K)=8$; the terms $q^{13}t^4$ and $q^{17}t^6$ are off the two diagonals $j-2r=7$ and $j-2r=9$, therefore $K$ is Kh-thick.




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Just for reference, I used the Java program, typing  

`java -classpath $CLASSPATH org.katlas.JavaKh.JavaKh < pd`  

(See page 29 [here](http://arxiv.org/abs/0906.5177) to set up the classpath correctly.)


The input file pd contains the [Planar Diagram](http://katlas.org/wiki/Planar_Diagrams) presentation, which in this case is  

`PD[X[1,5,2,4], X[3,7,4,6], X[5,3,6,2], X[7,19,8,18], X[8,14,9,13], X[11,1,12,22], X[12,18,13,17], X[15,11,16,10], X[16,22,17,21], X[19,15,20,14], X[20,10,21,9]]`


