I have such a cable attached to my house (with another unused one for phone, and an actually desired one for electricity). It broke free in a windy storm and was lying on the lawn. I carefully coiled it (a hundred feet of it or more, and flipped constantly to keep it from twist-induced kinking... did a nice job) around the base of the pole it came from. Done, nice, no longer an ugly thing hanging across the back yard, and hanging low enough that it pressed awareness of it being there upon one. Yay!
About a year later, the power company came through on their annual cut-away-tree limbs-from-being-dangerous-to-their-power-lines maintenance. After ten-ish years of never having heard from Time-Warner, nor ever having subscribed to anything from them, a couple days after the electric company came through, Time-Warner showed up and re-hung and re-attached the cable.
Thanks... actually never expected such service from them as their reputation and an out-house's are roughly the same. Kudos, sort of, and apologies to the local people. Not sure if the power company informs them of such things when noticed, or if they come along behind the tree cutting as being a good opportunity to do this (probably about 99.9999% of the time appreciated) work, but there it is.
So you could even go to the extreme of yanking it free from the house so that it looks storm-damaged, not viciously chopped down, or even nicely uninstalled, and coil it up by the pole and... probably you'll come home some day to find it nicely reinstalled, all part of a day's work for someone at Time-Warner.
So from my (limited) experience, they value the connections to homes. They did not even contact me first, just fixed it. It would be easy to see them being pretty unhappy for you to just whack it down, and even aggravate your life with a bill for it, the value of things they will now throw away, with said bill being sold to scum as a bad debt and them REALLY aggravating your life. In your area, it might even constitute a crime, that of vandalism, or some similar such. Which, even if not fully pursued could be a large stick used to get the bogus bill paid. I'd tread carefully, considering those things.
But don't just whack it, I think. Lots of (maybe serious) downsides and you might even come home to find it back in place so it just doesn't seem a good idea.