Has it flowered, it has white flowers....
The plants are fully grown by early summer and mature canes are hollow with a distinctive purple speckle and form dense stands up to 3 metres high. The plant flowers in late summer and these consist of clusters of spiky stems covered in tiny creamy-white flowers. - See more at: http://www.devon.gov.uk/index/environmentplanning/
natural_environment/biodiversity/japanese_
knotweed/identifyjapknot.htm#sthash.BXXDJ5TH.dpuf

If it is knotweed then you are required to remove it. Best to get an expert in because it's a so and so to get rid of.

Also look at http://www.environetuk.com/Japanese-Knotweed-Identification, they have a free identification service from photos. Yours does look possible, although the mature canes should be somewhere around - have you looked over the fence?

The leaves and the stem look suspiciously like it, although one would expect canes somewhere. I still wouldn't risk it. Buy some Rosate Green glyphosate (currently on special offer from Amazon) and zap it. You can kill brambles and trees with this stuff, without harming the soil and local environment

Looks like Russian vine, Mile-a-Minute (Falopia baldschuanica) to me. A picture showing where it comes out of the ground would be useful.

If it's M-a-M it will probably flower in the next few weeks (depending on how much sun it has had on it).

The problem with calling in 'experts' to get rid of JKW is that less than scrupulous people will tell you it's JKW to scare you into handing over a lot of money by telling you that it is illegal to have it in your garden. It is NOT illegal to have JKW in your garden but it is illegal to allow it spread to neighbouring land.