Why does this karyotype start numbering at 11?
Why does the band numbering start at 11? When I count other things I usually start with the number one.


It does start with the number one!

What is happening here is clearly a small confusion. Looking at the centromere in this idiogram of chromosome 12 you can see two number 11, as you stated in your question, one above and one below the centromere:



An idiogram of a Giemsa-stained chromosome 12. © 1994 David Adler, University of Washington. All rights reserved.

However, those two 11 below and above the centromere are not, in fact, an 11 (eleven): they are just two number 1 together, 1-1 or one-one. 

According to Nature (2017):


  This particular idiogram depicts the pattern of Giemsa staining at a fairly low resolution (i.e., it produces about 400 total bands in a karyotype, which is just above the threshold that is clinically useful). At this resolution, the long q arm of chromosome 12 can be subdivided into two main regions, which are designated 12q1 and 12q2. Region 12q1 can be further subdivided into five subregions, designated 12q11 through 12q15, each of which corresponds to a band detected by Giemsa staining. Orally, these subdivisions are referred to as "12q one-one" through "12q one-five" (not as "12q eleven" through "12q fifteen"). The more distal 12q2 region can be subdivided into subregions 12q21 through 12q24. In addition, subregion 12q24 can be further subdivided into regions 12q24.1 through 12q24.3, even at this relatively low resolution.. (emphasis mine)


Thus, those two 11 you see above and below the centromere are (or should be), respectively, 12p1-1 and 12q1-1 (remember that this is the chromosome 12).

The confusion here, therefore, is due to:


The omission of a 12p or a 12q before all those numbers.
The omission of some separator, like an hyphen or a comma, between those numbers.


A detailed PDF from the International System for Cytogenetic Nomenclature (ISCN) can be found here.

Bonus: In case you want to know, that number 850 in the top of the idiogram refers to the number of bands:


  At this higher level of resolution, approximately 850 bands can be distinguished in a karyotype


Source: Nature.com. (2017). Chromosome Mapping: Idiograms | Learn Science at Scitable. [online] Available at: https://www.nature.com/scitable/topicpage/chromosome-mapping-idiograms-302 [Accessed 23 May 2017].