2022/11/01 by Nikkuni, Ryo
#57K10 #57M15 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2211.00408
It is known that for every spatial complete graph on n≥ 7 vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to rn modulo (n-5)!, where rn = (n-5)!/2 if n=8k,8k+7, and 0 if n≠ 8k,8k+7. In particular the case of n=7 is famous as the Conway--Gordon K7 theorem. In this paper, conversely, we show that every integer (n-5)! q + rn is realized as the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots in some spatial complete graph on n vertices.