lol, believe me, I've only just looked into this and in the heat of the moment probably still bet it.
I know I only posted this a few hours ago, but we've already got tons of responses and I think we can talk about what the 'correct' answer should be: to
check, mostly for reasons already mentioned. This is what's called a
BA/WB situation (ie barely ahead/way behind). Similar to wa/wb in that we want to control the pot, but different in other ways.
One reason we're checking behind here is to keep the pot small; we don't want to build a big pot here since any hand that villain is putting in lots of chips with will likely either have us dominated (ie WB), or will be a hand that is drawing to significant outs (ie BA). If we wait until the turn to put a bet in, we a) take a lot of fold
equity away from big draws, and b) buy ourselves a lot more equity in the pot.
For example let's use QJs:
On flop:
equity win tie pots won pots tied
Hand 0: 37.121% 36.06% 01.06% 357 10.50 { AdJc }
Hand 1: 62.879% 61.82% 01.06% 612 10.50 { QsJs }
On turn:
equity win tie pots won pots tied
Hand 0: 59.091% 59.09% 00.00% 26 0.00 { AdJc }
Hand 1: 40.909% 40.91% 00.00% 18 0.00 { QsJs }
Now I'm no LHE guru, but there's a somewhat similar situation in SSHE where the author talks about waiting with TT until the turn to raise since the equity jump from flop to turn is so high. Just something to think about.
Regarding giving a free card, that definitely is a problem, but not as big of a problem as stacking (or building the pot for a stack) with TPTK here. We're essentially giving up a free card in exchange for keeping the pot small here, which is much more important.
Now what happens if we bet:
Villain checks, we bet ~$11, villain raises $35, we are feeling stuck.
For all we know, we could be up against a made hand, a big combo draw like Q♠J♠ or an outright bluff here. Now we have three very ugly looking options: fold what is possibly the best hand in what is now a pretty big pot, call and give him the advantage in the hand, and 3-bet with a hand we could be drawing near-dead with. None too appealing.
I think the default line should be to check behind, and usually bet when checked to on the turn, or call a turn bet if it's non-scary. By checking, we not only control the pot size, but induce bluffs as well. The river is where it might get tricky, but we'd deal with that when we got there by figuring out how likely villain would be to double barrel bluff based on stats etc.
) for protection with any typa hand we'd raise with pf. And to return to the beginning of this post, we are not losing any value from the hands we are currently raping by checking behind - in fact we're gaining value from them as they could 
