It would be better if you found or made one of those cables. I talked to one of our old-timers, and he just made his own from a standard phone cable and a db-9 connector, and the pinout chart in the manual. The AS3500 was made by Spectra-Physics, now a part of Thermo. The driver to use in PeakNet is TSP AS3500.
If you must use the relays or TTL, you use commands like "Pump.Relay_1.state = closed Duration = 1" or "Pump.TTL_1.on Duration = 1" to send signals to the AS3500 and "Wait Pump.Input_1 timeout=60" to wait for the TTL input. You need to make sure that the required relays and TTLs are enabled in the server configuration; you can rename them to make more sense in your programs. (Check my syntax, I'm doing this from memory.)