Cables of the figure-eight knot via real Frøyshov invariants

Key Points

• The aim is to show that the .2n;1/-cable of the figure-eight knot is not smoothly slice when n is odd.
• Proved non-smooth slice condition using real Seiberg-Witten Frøyshov invariant.
• Developed O.2/-equivariant version of the lattice homotopy type to aid computation.
• Computed real Seiberg-Witten Floer homotopy type for specific knots.
• Presented calculations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots.
• Confirmed that the .2n;1/-cable of the figure-eight knot is not smoothly slice for odd n.
• Developed new computational techniques for the real Seiberg-Witten Floer homotopy type.
• Provided new insights into the behavior of 2-knots through empirical results.