Periodic spanning surfaces of periodic knots

A knot in the 3-sphere S3 is called “periodic” if it is preserved by a finite order, orientation preserving diffeomorphism of S3, a property that can be viewed as an internal symmetry of the knot. Many knot invariants exhibit special patterns when applied to periodic knots, patterns that reflect said symmetry. 

In this talk we focus on genera of periodic knots, derived from various spanning surfaces of the knot. We shall review what is already known for the case of “periodic” orientable spanning surfaces in S3 and in the 4-ball D4, and then discuss novel results for the case of “periodic” non-orientable spanning surfaces, again, both in S3 and in D4. We shall compare and contrast the resulting “periodic knot genera” with their “non-periodic” brethren.