I have an s1642 lawn tractor with a Kohler Commander 16 engine, new battery, starter and solenoid. It won't turn over fast enough to start and both /- battery cables get very hot to the touch after trying. I have opened, cleaned and closed all connectors and ground bolted wires. Occasionally and rarely, it will kick over really easy, so I know all the parts work. But something is not working 95% of the time. Any ideas?

Try running a ground wire from the battery to the engine,
Sonds like its not getting a good ground ,a lot of these just have a grong going to the frame .

What did you find not helpful , the ground is the problem,

I ran a heavy jumper cable direct from the (-) battery to the engine block, but this made no difference. Turned out. Then I also replaced (8 to 6 gauge) the ground cable & solenoid to starter cables to be certain.
But thanks anyway for the lead, DG.

Okay uaually when the cables get hot this means its pulling too much current , which is from bad ground , but sounds like you got that covered , now on the starter did you replace it with a new starter ? Sometimes the bendix on the starter will bind and put the starter under a load .
Luck