Good invariant quantifying how complex a knot is

Question

I am currently trying to get upper bounds on the unknotting numbers of a list of knots. What I did is the following: I start with a diagram of the knot, then for each crossing, I simplify the knot which is obtained by changing the respective crossing. Then I check which crossing change reduces the amount of crossings of the knot the most. Then I repeat this with this new knot, until i get a knot of known unknotting number. The upper bound I get is then the number of crossing changes + the unknotting number of the final known knot. Now it turns out that this doesnt work well for my list of knots, as I see that my program only reduces the total number of crossings by 2 (or sometimes 3, but not often) each time I do a crossing change. Now my question: Is there some indicator or invariant which could tell me which knot obtained by a crossing change is the best to unknot the original knot the fastest way possible? (In my case, my indicator was the number of crossings, but this turned out to be not ideal...). I hope my question is clear. Thank you for your help.