We already defined a proper (or reduced) graph as a graph without loops. In the same way, a KL shadow is called proper, or reduced if it has no loops. A Gauss code {{1,2,3,4,2,3,4,1}} certainly represents a possible knot shadow (you can recognize a shadow of a trefoil with a loop). In an original KL (or, more precise, polygonal KL) we can easily get rid of the parts projected in a loop in the shadow, so we will do the same with loops in every shadow. In a Gauss code, the appearance of the same numbers in adjacent places indicates a loop (where the first and the last number in any component are treated as adjacent as well). 

From the Gauss code of a KL we can extract another, moore concise code- the Dowker code of the KL. In the case of knots, we only need to find the positions that the same numbers occupy in the Gauss code. For example, in the Gauss code of the figure-eight knot {{1,2,4,3,2,1,3,4}} the positions of the number 1 are {1,6}, the positions of the number 2 are {2,5}, the positions of the number 3 are {4,7}, and the positions of the number 4 are {3,8}. In every pair we put an odd number at the first position ({1,6}, {5,2} , {7,4}, {3,8}), and then sort the list of the ordered pairs obtained, so the result is D = {{1,6},{3,8},{5,2},{7,4}}. By reading every second member of the ordered pairs from the list D we obtain the Dowker code {{6,8,2,4}}. Because it has only one component and 4 crossings, we write it in the form Dow = {{4},{6,8,2,4}}, where the first part is the number of vertices of the component. 

According to the Jordan curve theorem (Jordan, 1887; Grabowski, 2005) saying that if c is a simple closed curve in ², then ²\c has two components (an "inside" and "outside"), with the c boundary of each, every crossing in Dowker code is denoted by one odd and one even number. The list D is sorted according to the first members of ordered pairs, this means, according to odd numbers. Hence, we don't need to work with the complete list, but only with a half of it. 

In the case of links everything works in the same way as with knots, except that sometimes it may happen that the correspondence "odd"-"even" is disturbed in some component. In that case we need first to rotate the component with the parity disturbed for one place, and continue in that manner till the parity is fixed for all components.