Trek Mamba 2013, what BB and crankset for 1x?

The only thing you need to know with respect to compatibility is what bottom bracket your frame can accommodate.

This old Bike Radar review says the Mamba has an ‘3 piece’ crank and Octalink cartridge BB as standard, which means the frame is a threaded shell type either 68 or 73mm wide.

Your easiest path forward is to get a ‘2 piece’ crank that uses external threaded BB cups and is compatible with 68 or 73mm frame shells. There are plenty of 1x cranks available that work with external bearings in the frame widths.

Note the BB cups have to match the crank axle, each manufacturer has their own standard for axle and bearing sizes. The threads in the frame are standardized though and known as ‘ISO’ or ‘English’ threads.

It’s not practical to cite a few suitable crank models because 1) product recommendations are off-topic here and 2) there are actually a large array of cranks from different manufacturers that will work. To start you off though:

• Shimano cranks use a 24mm axle and all work with their ‘Hollowtech II’ threaded BB cups (as well as press-fit BB types).

• SRAM offers cranks that are either 24/22mm axle that work with ‘GXP’ threaded cups, or 30mm axles that work with press-fit BBs.

If you want to understand Shimano’s products their product line-up charts are a really good place to start (Shimano is undoubtedly the best groupset manufacturer at providing documentation). This tells you what components work together and provides links to specifications.