Examples of topologically unknotted tori

We show that certain smooth tori with group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 4"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-surfaces in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 4"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 4"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> branched along any of these surfaces is a potentially exotic copy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S squared times upper S squared"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S^2 \times S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and, in the case of turned twisted tori, we show they cannot be distinguished from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S squared times upper S squared"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">S^2 \times S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using Seiberg–Witten invariants.