Y-DNA

The DNA test results can be viewed via the link 'DNA results' in the side menu. At present the option which provides the clearest result is the classic chart.
This shows that the two tests so far are from Go(a)thams with different haplotypes, meaning any link is likely to have been in pre-historic time, certainly well before the period covered by genealogical research.

The Kent Goatham has haplotype R-M269, the most common European haplogroup, and especially found to the West of Europe (Wikipedia article).
The German Gotham has haplotype I-M253, more common further East in Europe, especially in Scandinavia and Denmark (Wikipedia article; article with map).

The results will show subgroups according to results - so subgroups may be divided or merged, but the coloured headings below are those currently anticipated. 

English Go(a)thams

 Devon Gothams 

No results - none known to have tested yet.

Plan: initially recruit 2 men, one male-line descendant from each of the two main surviving groups, i.e. one a Gotham descendant of 'Abbotskerswell Agnes' and the other a descendant of 'Abbotskerswell Andrew'
** I have 2 DNA kits to test one man from each of these groups for free - please contact me if you are interested **
(n.b. Gotham-Potters from New Zealand do not count, as if you are one of them you descend from a female Gotham married to a male Potter, so should have Potter Y-DNA)

 Kent Goathams 

One result 

 Midlands Gothams 

(many Gothams living in Stoke-on-Trent from the C19th are descended from Devon Gothams; this DNA study subgroup is for those with earlier ancestry from Shropshire / Staffordshire)

No results - none known to have tested yet

USA Go(a)thams

English origin

No results - none known to have tested yet.

• English C19th- C20th ancestry
  Some descend from Go(a)thams who emigrated in the C19th or C20th; the ancestry suggested by documents is mostly clear. Benefits of a DNA test depend firstly on whether an English relation has tested (see above) and if they have then are quite limited - simply in confirming this link.
• English C18th ancestry
  No surviving documents have been found which make clear where 'John the emigrant' was from, though it seems probable that he was Kentish (a mariner John Gotham of a suitable age disappearing from the records) but quite possible he was from Devon (no likely John has been identified but there are known to be gaps in the surviving registers). ** descendants of John could learn a lot about their English ancestry by taking a Y-DNA test **  

 German origin 

One result. More results, especially from those descended from Gothams living in different parts of Germany, would be of interest.