# Double Slice Genus

Every knot in R^3 is the cross-section of an embedded surface F in R^4 where F has the property that it bounds an unknotted solid handlebody in R^4. For example, if G is a genus g Seifert surface for K, then G x [-1 , 1] is an unknotted solid handlebody in R^4 with boundary as desired. Its boundary is of genus 2g.

The * double slice genus of K, g*_{ds}(K) is the minumum genus among all such surfaces F. In the case that g_{ds}(K) = 0, K is called * doubly slice*. Clearly 0 ≤ g_{ds}(K) ≤ 2g_{3}(K). References [2,3] provide background and initial work studying the double slice genus. The most extensive work and the data here was provided by Karageorghis and Swenton, published in [1].

Knots with double slice genus 0 are called doubly slice. There is extensive literature on this topic which we do not cite here.

## Specific Knots

Here we list any specific examples that need reference. Usually these appear in the results page with the values linked to this page.

12a_{{153}}, 12n_{{239,512}}.

Ref. [4]

## References

Here we post some references.

[1] Karageorghis, C. and Swenton, F., "Determining the doubly slice genera of prime knots with up to 12 crossings." Arxiv preprint.

[2] Livingston, C. and Meier, J., "Doubly slice knots with low crossing number," New York Journal of Math., 21 (2015), 1007-1026.

[3] McDonald, C., "Band number and the double slice genus," New York Journal of Math., 25 (2019), 964-974.

[4] Brittetnham, M., Hermiller, S., "The smooth 4-genus of (the rest of) the prime knots through 12 crossings ," Arxiv preprint.