We know that a sphere has topological genus 0, and if we punch a hole in the sphere, we get a donut with genus 1, and we increase the genus by 1 for each additional hole we punch. So the "punch a hole" operation increases genus by 1, ergo the inverse "fill a hole" operation, like filling the hole in a donut to get a sphere, decreases genus by 1. Likewise, if we start with the donut, and slice through one side, it becomes topologically equivalent to the sphere again, so the "cut apart" operation decreases genus by 1 (given that our cut has a simple closed curve as a profile, that is, slices through a single solid piece, not through any holes or bubbles in the object). Likewise, the inverse operation of "join together" increases genus by 1. If we stretch a sphere of genus 0 into a rope and join the ends into a donut, we increase the genus by 1. In fact, "punch a hole" and "join together" perform the topological operation "add a handle", while "fill a hole" and "cut apart" perform "remove a handle" ... supposing a handle exists. What happens if we cut a sphere into two pieces? A single sphere has genus 0, the "cut apart" operation should therefore produce two spheres together having genus -1. If we cut one of those spheres, we end up with 3 spheres having genus -2. By induction, n spheres together have genus 1-n. The consistent conclusion is that 0 spheres, that is, empty space, has genus 1. Again, start with a sphere, this time flatten it into a sheet and blow it up like a balloon, it still has genus 0. Now use the "fill a hole" operation to close the opening of the balloon, producing a hollow sphere. Since "fill a hole" reduces genus by 1, the hollow sphere, or "sphere with a bubble", has genus -1. You can shrink the bubble, and repeat the operation, blowing another bubble in the sphere, closing the hole, and getting a sphere with two bubbles and genus -2. By induction, a sphere with n bubbles has genus -n. Now consider that n spheres have genus 1-n, while a sphere with n bubbles, that is, n spherical holes carved out of it, has genus -n. This suggests that complementing a figure with respect to an enclosing sphere decreases its genus by 1 (leastwise, we have an example for every negative n). We will call this the "hollowing out" operation, and its inverse would be the "make a cast" operation. So let's start with nothing, which we agree has genus 1. We hollow this nothing out of a sphere, which decrements the genus and produces a sphere of genus 0 (consistent so far). We hollow this sphere out of another sphere, producing a hollow sphere of genus -1, in agreement with our earlier assessment. We hollow this hollow sphere out of a yet bigger sphere, and we get a hollow sphere with a sphere inside, of genus -2. If we take a hollow sphere of genus -1, and pull a little knob out from its inside surface, and cut the knob off ("cut apart"), we likewise obtain a hollow sphere with a sphere inside of genus -2, so everything seems to be consistent. So what is the genus of the Borromean rings? Well, these are three intertwined donuts. If we cut each of the 3 donuts, the genus is reduced by 3, the rings fall apart, each donut becomes a sphere, and we get 3 spheres with genus -2. This means that the Borromean rings had genus 1. In fact, this same argument goes through for any configuration of 3 links, no matter how knotted. One the 3 links are cut, the genus is reduced by 3, the links become 3 spheres of genus -2, so the original links together had genus 1. Indeed, this same argument goes through for any number of links, no matter how knotted. Suppose there are n links. Once the n links are cut, the genus is reduced by n, the knot falls apart, and the n links become n spheres, togther with genus 1-n. This means that the original set of links had genus 1, that is, every knot or linkage has genus 1, including the unknot (donut) and the empty knot (empty space). You can also start with two donuts, each of genus 1. If you join them together, you increase their genus by 1, and end up with a two-hole donut, of genus 2. This means that the two donuts originally must have had genus 1. In fact, you can join n donuts together with n-1 joins, increasing their genus by n-1, to get an n-hole donut of genus n, meaning that all n donuts together had genus 1. This just verifies that any number of links together have genus 1. We can also surmise that the "juxtapose" operation of considering an object of genus n together with an object of genus m as a single object seems to yield an object of genus n+m-1. For example, a donut has genus 1, so two donuts together have genus 1+1-1 = 1. Juxtaposing this 2-donut object with a third donut again gives genus 1+1-1 = 1, and again we confirm our theory about knots and linkages all having genus 1. Likewise, a sphere juxtaposed with a hollow sphere has genus 0 + -1 - 1 = -2, as we earlier determined for a sphere inside a hollow sphere. Juxtaposing any object of genus n with empty space should result in the original object with the same genus n, confirmed by juxtaposed genus n+1-1 = n. So far, I haven't found any inconsistencies in the concept of negative genus. Supposing it is consistent, I must assume it is well known to the topology crowd. At any rate, I have heard it said that a topologist is someone who can't tell a donut from a cup of coffee. My condition is worse, I can't tell a dozen donuts from a cup of coffee.