Tilings of hexagons with a removed triad of bowties

In this paper we consider arbitrary hexagons on the triangular lattice with three arbitrary bowtie-shaped holes, whose centers form an equilateral triangle. The number of lozenge tilings of such general regions is not expected — and indeed is not — given by a simple product formula. However, when considering a certain natural normalized counterpart R of any such region R, we prove that the ratio between the number of tilings of R and the number of tilings of R is given by a simple, conceptual product formula. Several seemingly unrelated previous results from the literature — including Lai's formula for hexagons with three dents and Ciucu and Krattenthaler's formula for hexagons with a removed shamrock — follow as immediate special cases of our result.