If you've seen a Hematologist
and he/she has recommended Lovenox, then you've got a hypercoagulable disorder that makes you prone to blood clots
. Not being on an anticoagulant
greatly increases your risk of stroke, loss of a limb or death.
I am a bit surprised that warfarin (an oral anti-coagulant) isn't suggested, but the hematologist is the expert on clotting problems and I'd go with the recommendation.
You varicose veins
likely had little to do with the problem, as they are superficial, not deep veins. However, clots in the superficial veins can, over time,cause varicose veins.