Is it possible to determine programmatically what jack on a network switch a computer is connected to?
It's either possible or not possible, depending on the switch.
If it's a managed switch, you could query its management interface to determine the MAC addresses behind each port. Then you could match that up with the hardware address on the computer you're interested in.
Some switches have discovery protocols (such as CDP or LLDP) which will broadcast the port's identity to whichever devices are attached. You could sniff the network for these packets to determine the port your computer is connected to.