ABSTRACT
In this chapter, the authors use different techniques to construct knot and link diagrams. They use these techniques to create knots and links that meet specific criteria. In some cases, the diagrams are altered in a way that is not detectable by the invariants. The authors also use symmetry to create a family of link diagrams. They then apply a linking invariant to these diagrams to explore whether or not these new diagrams are equivalent. These examples suggest several theorems about the crossing weight numbers and diagrams obtained using symmetry. This process illustrates how mathematicians move from examples and computation to hypothesis and then finally to proof and theorems. The authors also introduce some additional ways to construct knots. They provide a version of Conway notation for virtual tangles that was developed by Slavik Jablan for use with the Mathematica program LinKnot.
