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.
A minimum 1% concentration of vitamin K1 has been shown in animal studies to positively influence wound healing by hastening the natural process damaged skin undergoes as it works to repair itself. It is not known if this principle would apply to intact skin showing signs of aging.
The same concentration of vitamin K1 was used in another study to test its results on improving the look of dark circles. Although some improvement was noted, the formula in testing also contained caffeine and emu oil, although the study’s authors concluded that the dark circle benefits were solely attributed to vitamin K.
Interestingly, despite somewhat encouraging research, we rarely see products that contain (or are likely to contain) the amounts of vitamin K research has shown can be beneficial.
References for this information:
Advanced Biomedical Research, January 2015, ePublication
Indian Journal of Pharmacology, July-August 2014, pages 409-412
Journal of Cutaneous and Aesthetic Surgery, July-September 2012, pages 176-182