# What is the meaning of $(h^{11},h^{21})\to (h^{11}-240,h^{21}+240)$ in Calabi-Yau threefolds?

By browsing through the Hodge data of known Calabi-Yau threefolds, I stumbled upon an observation that frequently enough a pair of Hodge numbers $(h^{11},h^{21})$ comes together with the pair $(h^{11}-240,h^{21}+240)$. This shift, together with mirror symmetry accounts for a certain symmetry in the top portion of the Kreuzer-Skarke plot.

Has anyone observed this shift before?

Is there an empirical explanation for this shift? Is there any relation to the roots of $E_8$?

Are there other similar but smaller shifts?

• A very interesting observation. Mar 22, 2014 at 1:00
• In the sactter plot, do the two parabola-like upper curves match up if one is translated to the other? Mar 22, 2014 at 7:55
• I believe so. I zoomed in on the picture near the vertex of the parabola and it was completely symmetric. Also notice that the top points are $(11,491),(251,251),(491,11)$ which exhibits the same shift twice. Mar 22, 2014 at 14:36
• Certainly very intriguing. I didn't realise that the curve was symmetric. Mar 22, 2014 at 17:45
• Look at this image: benjaminjurke.net/dynamic/cyexp/… There is almost a perfect symmetry in the top ranges, as if there was some sort of mechanism to get the new Hodge pair if the original numbers are large enough. Mar 22, 2014 at 22:00