Heegaard's ideas allowed him to make a careful critique of work of Henri Poincaré.

He later co-authored, with Max Dehn, a foundational article on combinatorial topology, in the form of an encyclopedia entry.

An English translation of his 1898 thesis, which laid a rigorous topological foundation for modern knot theory, may be found at https://www.maths.ed.ac.uk/~v1ranick/papers/heegaardenglish.pdf.

The section on "a visually transparent representation of the complex points of an algebraic surface" is especially important.

Following a talk titled "A Biologist's View of the Future of the White Race," Heegaard hosted a garden party for the Congress participants.