Estimating complexities of knotted objects from diagrams

Advances in topology enable us to study knotted objects by drawing pictures on a piece of paper. It can be difficult to guess how complex a knotted object is from looking at its diagram. For instance, most tabulations of knotted objects are done in terms of number of crossings. However, a crossing number minimizing diagram usually does not minimize the number of local maxima the knotted objects need. In this talk, I will discuss some methods that tell us when knotted objects admit (possibly different) diagrams with lower complexities.