"A line that divides the mass in half is called the center of mass."
Actually that is not correct. "Center of mass" has a definition...
"The center of mass is the point where all of the mass of the object is concentrated. When an object is supported at its center of mass there is no net torque acting on the body and it will remain in static equilibrium."http://dev.physicslab.org/Document.aspx?doctype=3&filename=RotaryMotion_CenterMass.xml
That is why I use my term "Center of Mass
Line"...there is a difference.
We all know how the energy of a hammer blow can transfer to the wrist if the blow is off center or the handle strikes the object instead of the head. In a big hammer or axe, the transferred energy can be enough to break your wrist bones. In swords, it can be nearly as dramatic. On a knife it is more subtle.
It is pretty easy to visualize how moving the center of mass affects how a knife cuts. If you replaced the broom with a knife in AVigil's drawing, it would shows that at the center of mass, the knife is "balanced". We often say, "The knife/sword has a balance point just at the ricasso", etc. This means that there will be no torque applied to the knife if you place your finger at the ricasso on one side...the knife will balance. If you apply force to either end ( handle or blade) an opposite movement would happen at the other. Moving the mass forward or backward by making the blade and handle lighter or heavier moves the balance point. That is pretty basic and merely a teeter-totter view. However, this is sort of a 2 plane balance...up and down. or left and right. X and Y as it is called in engineering.
OK, this is fine as long as the piece being balanced is basically a straight line. We all see how a bowie or a rapier feels better or worse if the center of mass is moved.
But, if you look at the blade in profile, and imagine the balance point in true 3D as not just a fulcrum point, but add a line dividing the mass of both X and Y along a Z axis, we add a big change to the term "balance". Now the blade can rotate along the tip to butt line ( rotate on the Z axis) in response to changes along the X/Y planes.If the blade edge is far below the centerline, as force is applied to the edge it will apply a torque to the center of mass point. Your grip on the handle prevents the tip from moving along the X and Y planes, so the blade will try and rotate on the Z axis. In the 2D view, it just deflected up/down or left/right. In the 3D view, it can rotate on the centerline ( twist in the hand). If it was a frictionless situation, it would rotate until the edge was pointing up and the spine was trying to cut. Since there is an edge biting into the cutting object and your hand gripping the handle, the blade does not rotate....but the energy applied to the edge tries to rotate. Thus the edge that is much too low will try and rotate against your grip, which overpowers the rotation in most cases. That transfers part of the energy of the cut back into your hand. If the edge was near the centerline, or above it, all the energy would be transferred to the edge and the cut would have more "power". How a knife "cuts" and "handles" is a factor of these things. Moving the edge and tip along these planes can make a knife function differently. The ways to change this situation are to either apply much more mass below the line ( kukri); or reduce the total mass to a point where the hand easily overpowers the applied torque ( light weight knives); or apply the force along a longer plane on one axis only ( 2" tall blade on a chef's knife); or raise the tip above the line and make it have much more control ( fillet and skinner).
A well planned slicing knife, like a yanagi-ba, can be held almost with just two fingers and slice like a laser. It's handle is thin and often just a round stick. Since the blade centerline is on the same line as the handle, it reacts to every move of the hand with almost no resistance. The lower the mass of the blade, the higher degree of response to the handle movements.
A big chopper or kukri with a dropped edge for transfer of energy will have a big grip and finger grooves to assure the force applied by the moving hand and arm are transferred as straight as possible to the edge to drive it deep into the target object. The higher the mass of the blade, the more the impact drives the edge in.
This was sort of fast and not heavily proof read, but I hope it explains it a bit better.