There are the predictable issues facing the candidates running for mayor of New York: jobs, education, crime.

And then there is circumcision.

In a classic of the “only in New York” genre, both major candidates have found themselves tripped up by an arcane debate pitting public health against religious freedom, prompted by the city’s efforts to regulate what it views as a dangerous variation on the traditional ritual in some corners of the ultra-Orthodox Jewish community.

The Democratic candidate, Bill de Blasio, and the Republican candidate, Joseph J. Lhota, have both pledged to revisit the issue if elected as mayor. But their stances have raised questions because, in Mr. de Blasio’s case, he stood silently as Jewish leaders described his position in different language than he has used, and, in Mr. Lhota’s case, he has changed his position over the course of the campaign.

At issue is a practice called, in Hebrew, metzitzah b’peh, in which the officiant performing the circumcision, who is called a mohel, uses his mouth to remove blood from an infant’s circumcision wound.