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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.

Author Bio

Megan Gregory is an undergraduate student at Massey University in Palmerston North, New Zealand. She is majoring in mathematics, and plans to pursue a postgraduate degree in maths after completing her undergraduate studies. The paper, "Proving n-Dimensional Linking in Complete n-Complexes in (2n+1)-Dimensional Space" was written during the summer of 2009-2010 as part of the Institute of Fundamental Sciences Summer Scholarship program with editing continuing into 2011. Megan worked on this paper with Dr. Christopher Tuffley of Massey University who was acting as her supervisor. Megan would be happy to hear from anybody with questions about her work or for discussions about mathematics in general, she can be reached via email.

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