We establish a connection between the theory of Ulrich sheaves and A¹-homotopy theory. For instance, we prove that the A¹-degree of a morphism between projective varieties, that is relatively oriented by an Ulrich sheaf, is constant on the target even when it is not A¹-chain connected or A¹-connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the A¹-degree of all its linear projections can be read off 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 to define an arithmetic version of Viro’s encomplexed writhe for curves in P³. This can be considered to be an arithmetic analogue of a knot invariant. Namely, we define a notion of algebraic isotopy under which the arithmetic writhe is invariant. For rational curves of degree at most four in P³ we obtain a complete classification up to algebraic isotopies.