Technically known as phytonadione, vitamin K is a fat-soluble vitamin. Applying vitamin K to the surface of the skin won't improve the look of unevenness, or dark circles. This type of vitamin K is also referred to as vitamin K1.
A study published in the Journal of Cosmetic Dermatology (April 2004, page 73) examined the effect of applying a gel containing 2% vitamin K plus 0.1% retinol, vitamin E, and vitamin C. Fifty-seven adults with dark circles participated in this 8-week study and the results, while not a slam-dunk, weren't exactly discouraging either: 47% of the testers noted "fair to moderate" improvement in their dark circles. The majority of testers noticed no change, but the ingredient was well-tolerated. As encouraging as this seems, whether or not the results were from the vitamin K or the other vitamins is unknown.