New York Journal of Mathematics

New York J. Math. 14 (2008) 285–287.

Corrigendum to “Knotted Hamiltonian cycles in spatial embeddings of complete

graphs”

Joel Foisy

Abstract. We state and prove a correct version of a theorem presented in Blain, Bowlin, Foisy, Hendricks and LaCombe, 2007.

Professor Masakazu Teragaito has recently pointed out that Theorem 3.3 of [1] is incorrect as stated. The fact that μ_{f}(G, Γ; 6) = 3 is independent of embedding of K_{8} does not necessarily imply that there are at least 3 knotted Hamiltonian cycles. For example, there could be exactly one knotted Hamiltonian cycle with a_{2}(K) = 3.

Professor Kouki Taniyama has further pointed out that Lemma 2, in [2], has a gap in its proof. He has proposed a rigorous proof of a weaker version Lemma 2. In this short paper, we will state and prove this weaker version of Lemma 2 in [2], and then apply it to obtain a weaker version of Theorem 3.3 of [1]. For definitions of terms, see [2] and [1]. Here is the modified version of Shimabara’s Lemma 2 that we will prove:

Lemma 1. Let Γ be a set of cycles in an undirected graph G. The invariant μ_{f}(G, Γ; n) does not depend on the spatial embedding f of G if the following two conditions hold:

(1) For any edges A, B, E such that A is adjacent to B,

ν_{1}(Γ; A, B, E) ≡ 0 mod n.

(2) For any pairs of nonadjacent edges (A, B) and (E, F),

ν_{2}(Γ; A, B; E, F) ≡ 0 mod 2n.

The difference between this new lemma and the original comes in the second condition, where the equivalence is mod 2n, not n. For the proof of Lemma 2, case 2, on p. 410 of [2], the definition of linking number used does

285^{ISSN 1076-9803/08}

Received July 10, 2008.

Mathematics Subject Classification. 57M15, 57M25.

Key words and phrases. spatial graph, embedded graph, intrinsically knotted.

not work. For the version of linking number used, ζ(A, B) depends on the order of A and B, whereas the equality

∑_{E,F} ∑_{γ∈Γ_{1}} ϵ(c)ζ(f_{γ}(E), f_{γ}(F)) = ∑_{E,F}(n_{3} − n_{4})ζ(f(E), f(F))

implicitly uses the assumption that ζ(A, B) = ζ(B, A).

A proof of Lemma 1 is possible if one uses a different (but equivalent) version of ζ and linking number. For A and B two disjoint oriented arcs or circles in ℝ^{3}, define

ζ′(A, B) = \frac{1}{2} ∑_{c} ϵ(c),

with the summation being over all crossings between A and B. If A and B are circles, then ζ′(A, B) gives the linking number of A and B, lk(A, B). It then follows immediately that ζ′(A, B) = ζ′(B, A). The proof of Lemma 1 proceeds as in the proof of Lemma 2 in [2], but now ends with:

δ(u) = ∑_{E,F} ∑_{γ∈Γ_{1}} ϵ(c)ζ(f_{γ}(E), f_{γ}(F)) = ∑_{E,F}(n_{3} − n_{4})ζ′(f(E), f(F)).

It then follows, since each ζ′(f(E), f(F)) is either an integer or an integer divided by 2, that, δ(u) is congruent to 0 mod n if |n_{3} − n_{4}| ≡ 0 mod 2n.

Now, in [1], it was shown that ν_{2} ≡ 0 mod 6. It was also shown that there exists an embedding of K_{8} with exactly 21 knotted Hamiltonian cycles, each with Arf invariant 1. One can also verify that each of these knotted cycles is a trefoil, with a_{2} = 1. This embedding, together with Lemma 1, implies that μ_{f}(K_{8}, Γ, 3) ≡ 0 for every spatial embedding of K_{8}. By Theorem 2.2 of [1], there is at least one Hamiltonian cycle with Arf invariant 1, in every spatial embedding of K_{8}.

We thus have the following corrected version of Theorem 3.3 from [1].

Theorem 2. Given an embedding of K_{8}, at least one of the following must occur in that embedding:

(1) At least 3 knotted Hamiltonian cycles.

(2) Exactly 2 knotted Hamiltonian cycles C_{1} and C_{2}, with

a_{2}(C_{1}) ≡ 1 mod 3 and a_{2}(C_{2}) ≡ 2 mod 3,

or

a_{2}(C_{1}) ≡ 0 mod 3 and a_{2}(C_{2}) ≡ 0 mod 3.

Either C_{1} or C_{2} has nonzero Arf invariant.

(3) Exactly 1 knotted Hamiltonian cycle, C with

a_{2}(C) ≡ 1 mod 2 and a_{2}(C) ≡ 0 mod 3.

(Equivalently: a_{2}(C) ≡ 3 mod 6.)

Corrigendum to Knotted Hamiltonian cycles 287

It thus remains an open question to determine if 1 is the best lower bound for the minimum number of knotted Hamiltonian cycles in every spatial embedding of K_{8}.

Acknowlegments The author would like to thank Professors Masakazu Teragaito, Kouki Taniyama and Ryo Nikkuni for valuable comments and suggestions.

References

[1] Blain, Paul; Bowlin, Garry; Foisy, Joel; Hendricks, Jacob; LaCombe, Jason. Knotted Hamiltonian cycles in spatial embeddings of complete graphs. New York J. Math. 13 (2007) 11–16. MR2288079 (2008a:05061).

[2] Shimabara, Miki. Knots in certain spatial graphs. Tokyo J. Math. 11 (1988) 404– 413. MR0976575 (89k:57018), Zbl 0669.57001.

Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676 foisyjs@potsdam.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-13.html.