Without a dose response, I'm inclined to believe that the increase in lymphoma seen in people with tattoos has more to do with confounding factors than with the ink or the act of getting a needle poked into your skin. I would think that controlling for all confounders in a study like this would be exceptionally difficult.
That said, I'm pretty sure that at least some inks do contain known carcinogens[1]
"Sharing traits common in motor cycle gangs increases risk of cancer - new study"
For those who don't know how to interpret medical evidence, this study is very weak.
"The mean age of death for tattooed persons was 39 years, compared with 53 years for non-tattooed persons (P = .0001). There was a significant contribution of negative messages in tattoos associated with non-natural death (P = .0088) but not with natural death."
TFA has a more direct, physical, concern - it starts from a well-known, that tattoo ink ends up in lymph nodes, and it does a statistical analysis showing there's a significant statistical result in lymphoma occurence.
I think people with negative tattoos dying younger reduces the # of people with tattoos who get lymphoma, as they have less ink-in-lymph-nodes years.
I doubt they intended to communicate something that general, and if they did, I doubt they meant to pick one that would reinforce the conclusion.
Also the slicing and dicing, “11 more than the index year” and so on, is multiple hypothesis testing on the face of it; I wonder if they adjust for that.