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We establish a connection between the theory of Ulrich sheaves and$\mathbb{A}^1$-homotopy theory. For instance, we prove that the$\mathbb{A}^1$-degree of a morphism between projective varieties, that isrelatively oriented by an Ulrich sheaf, is constant on the target even when itis not $\mathbb{A}^1$-chain connected or $\mathbb{A}^1$-connected. Further ifan embedded projective variety is the support of a symmetric Ulrich sheaf ofrank one, the $\mathbb{A}^1$-degree of all its linear projections can be readoff in an explicit way from the free resolution of the Ulrich sheaf. Finally,we construct an Ulrich sheaf on the secant variety of a curve and use this todefine an arithmetic version of Viro's encomplexed writhe for curves in$\mathbb{P}^3$. This can be considered to be an arithmetic analogue of a knotinvariant. Namely, we define a notion of algebraic isotopy under which thearithmetic writhe is invariant. For rational curves of degree at most four in$\mathbb{P}^3$ we obtain a complete classification up to algebraic isotopies.

Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves