Chuck norris can play hotel california on a triangle... Jus' sayin'
For 'cut' on LPs I always suggest 500k pots, 50s wiring and pickups with alnico 5 magnets, isometrically wound and of moderate to low output. Before I changed out the Wilkinsons on my two V100s they were muddy as a warthog's bollocks, so if your Les Paul is less defined than one of those you may have a problem! I'd try pots/fifties wiring first as they're probably the simplest option.
If writing doesn't do the job I would recommend Tonerider Alnico IV Classics HBs
They are the brightest and clearest HBs I've ever heard.
500K pots will help but I'd check the wiring for poor joins, especially on the pickup selector switch, I have a Les Paul that always sounded muddy on the neck pickup until I resloldered the tags on the switch. Sloppy work from the Gibson Custom shop.
i will have to take a look at the switch, it does cut out on the mod setting sometimes
I've got an LP Studio Deluxe 60s and there;s nothing muddy about the sound. So it may be worth having the guitar checked out for faults before you spend more money on new pickups.