Since NURBS curves are the projections of 4D Bspline curves to 3D, knot insertion for NURBS curves is easy. In fact, knot insertion for NURBS curves is done in three steps: (1) converting the given NURBS curve in 3D to a Bspline curve in 4D, (2) performing knot insertion to this four dimensional Bspline curve, and (3) projecting the new set of control points back to 3D to form the the new set of control points for the given NURBS curve.
Suppose we have n + 1 control points P_{0}, P_{1}, ..., P_{n} with associated weights w_{0}, w_{1}, ..., w_{n}, a knot vector U and a degree p. Let P_{i} = (x_{i}, y_{i}, z_{i}). Then, control points P^{w}_{i} = ( w_{i}x_{i}, w_{i}y_{i}, w_{i}z_{i}, w_{i} ), 0 <= i <= n, and knot vector U define a four dimensional Bspline of degree p. The insertion of a new knot t to this four dimensional Bspline yields a new set of control points Q^{w}_{i} = ( X_{i}, Y_{i}, Z_{i}, W_{i} ), 0 <= i <= n. Projecting them back to the threedimensional space by dividing the first three components with the fourth yields the new set of control points of the given NURBS curve.
Let us take a look at an example. Suppose we have 9 knots
u_{0} to u_{3}  u_{4}  u_{5} to u_{8} 
0  0.5  1 
and a NURBS curve of degree 3 defined by the following 5 control points in the xyplane:

x  y  w 
P_{0}  70  76  1 
P_{1}  70  75  0.5 
P_{2}  74  75  4 
P_{3}  74  77  5 
P_{4}  40  76  1 
The following shows the curve and its basis functions.
Let us insert a new knot t = 0.4. Since t is in knot span [u_{3}, u_{4}) and the degree of the NURBS curve is 3, the affected control points are P_{3}, P_{2}, P_{1} and P_{0}. Since this is a NURBS curve, we shall use homogeneous coordinates by multiplying all control points with their corresponding weights. Call these new control points P^{w}_{i}:

x  y  w 
P^{w}_{0}  70  76  1 
P^{w}_{1}  35  37.5  0.5 
P^{w}_{2}  296  300  4 
P^{w}_{3}  370  385  5 
Note that since P_{4} is not affected, it is not computed in the above table. Then, we shall compute a_{3}, a_{2} and a_{1} as follows:
The new control points Q^{w}_{3}, Q^{w}_{2} and Q^{w}_{1} are:
Projecting these control points back to 2D by dividing the first two components with the third (the weight), we have
The following is the resulting NURBS curve and its basis functions: