Proving n-Dimensional Linking in Complete n-Complexes in (2n + 1)-Dimensional Space

Abstract. This paper proves that there is an intrinsic link in complete n-complexes on (2n + 4)-vertices for n = 1, 2, 3 using the method of Conway and Gordon from their 1983 paper. The argument uses the sum of the linking number mod 2 of each pair of disjoint n-spheres contained in the n-complex as an invariant. We show that crossing changes do not affect the value of this invariant. We assert that ambient isotopies and crossing changes suffice to change any specific embedding to any other specific embedding. To complete the proof the invariant is evaluated on a specific embedding. Conway and Gordon use a diagram to carry out the final step for a 3-dimensional example and we use a computer to do this in higher dimensions. Our code is written in MATLAB. Taniyama has a proof for higher dimensions that does not use a computer. RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 19