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This way, we obtain reduced knot projections with the minimal number of crossings. Two such projections or knot diagrams are equal if they are isotopic in projection plane as graphs, where the isotopy is required to respect overcrossing respectively undercrossing. In order to classify our curves, treated as knot projections, we will define an invariant of knot (or link) projections.
Let be given a reduced oriented knot diagram D with generators g_{1}, ..., g_{n}. If the generators g_{i}, g_{j}, g_{k} are related as in the left figure, then a_{ii}=t, a_{ij}=1, a_{ik}=1; if they are related as in the right figure, then a_{ii}=t, a_{ij}=1, a_{ik}=1; in all the other cases a_{ij}=0. The determinant d(t)=a_{ij} is the polynomial invariant of D.
For example, for the oriented diagram of the knot 5_{2}, d(t)=t^{5}t^{3}5t.
The writhe of D, denoted by w(D), is the sum of signs of all the crossing points in D, where the sign is +1 if the crossing point is "left", and 1 if it is "right". It is the visible property of every knot projection: w(D) is the type of the knot projection.
There are some important properties of the integer polynomial invariant d(t)=c_{n}t^{n}+... +c_{1}t:
Let us also notice that this polynomial projection invariant makes distinction not only between nonisomorphic knot projections of prime knots (e.g. two projections of the knot 7_{5}, to which correspond, respectively, the polynomials t^{7}+3t^{5}4t^{3}7t and t^{7}+2t^{5}+t^{4}4t^{3}7t, but also between nonisomorphic knot projections of composite knots (e.g. three nonisomorphic projections of 4_{1}#3_{1}, with their projection polynomials t^{7}+t^{5}2t^{3}+5t^{2}3t, t^{7}+t^{4}3t^{3}2t^{2}+3t, t^{7}+t^{5}+2t^{4}2t^{3}+t^{2}3t, respectively).
Rectangular square grid RG[2,2] is the minimal RG from which we could derive some nontrivial alternating knot (different from unknot)  the knot 3_{1}. From RG[3,2] we could obtain 7_{4}, 6_{2}, 3_{1}^{+}#3_{1}^{+}, 5_{1}, 5_{2}, 4_{1} and 3_{1}, where different mirrorarrangements could give the same projection.
Could you derive every knot projection from some RG? Which knot projections could be obtained from a particular RG? Which mirrorarrangements in some RG result in the same knot projection? Find the minimal RG for a given knot! Could you obtain nonisomorphic projections of some knot from the same RG? These and many other problems connected with mirrorcurves represent an open field for research.