You need the 3 hoses that go to the sensor. They need to be equal length. The 3 hoses from the TBs go to a 4 way connector then a larger hose goes from the 4 way connector to the sensor. You need this for the engine to run.
Hoses: These hoses are critical on the Rocket 3 and the bike will never run (properly) without them. Just install them as seen in the photo, or if you don't have them, I suggest ordering them from Triumph. It's been a while since I've messed with them, but I seem to recall them being 3 different lengths (an important detail). If you can't wait, you can make your own hoses using the factory lengths from any type of fuel-safe vacuum hose.
ThrottlesInjectors amp Fuel Rail
14: T1242614 | Hose ID 3 Map Cyl 3 $9
15: T1242615 | Hose ID 3 Map Cyl 2 $9
16: T1242616 | Hose ID 3 Map Cyl 1 $9
The Rocket doesn't have a cam position sensor, instead, it uses vacuum pulse timing from each of these hoses to figure out when cylinder 1 is on the compression stroke, which is why these need to be the factory length sections of the hose. The timing delay caused by the known length of the hose section is how the ECU knows which vacuum pulse belongs to which cylinder.
Plugs: They're like $2.50 each or something at any autoparts store, just buy the cheap copper plugs. The Rocket works better with cheap plugs than expensive ones...odd, but true, it's a good problem to have
The plugs you've pulled out look physically fine, just covered in soot. If necessary, just wire brush the ground strap and electrode clean and reinstall.
DPR7-EA9 or DPR8-EA9 for the part number. The rocket doesn't care which you use, grab whichever is available more quickly or is cheaper.