It is not hard to cut a bagel into
two equal halves which are linked like two links of a chain.

To start, you must visualize four key points.  Center the
bagel at the origin, circling the Z axis. 
A is the highest point above the +X axis.  B is where the +Y
axis enters the bagel.
C is the lowest point below the -X axis.  D is where the -Y
axis exits the bagel.

These sharpie markings on the bagel are just to help visualize the
geometry
and the points.  You don’t need to actually write on the bagel to
cut it properly.

The line ABCDA, which goes smoothly through all four key points,
is the cut line. 
As it goes 360 degrees around the Z axis, it also goes 360 degrees
around the bagel.

The red line is like the black line but is rotated 180 degrees
(around Z or through the hole).
An ideal knife could enter on the black line and come out exactly
opposite, on the red line.
But in practice, it is easier to cut in halfway on