We introduce geometric formulas for several low degree knot and link invariants, expressing them as signed counts of certain quadrisecants, i.e., lines meeting the link in four points. We start by describing a simple unified construction using maps of configuration spaces and elementary intersection theory. By building a compactification modeled on the Fulton–MacPherson construction, we interpret these invariants directly as certain intersection numbers. We then switch to an intriguing geometric interpretation for signs of quadrisecants. It turns out that the signs are determined by cross-ratios of the four intersection points and their tangent lines (identifying the locus of sign changes as certain hyperboloids of one sheet). If time permits, in the end we will briefly discuss a graph complex governing these formulas and conjectural formulas for higher degree invariants Advisor: Prof. Michael Polyak