[Paper Review] Intrinsically knotted graphs with 21 edges

Jamison Barsotti, Thomas W. Mattman|arXiv (Cornell University)|Mar 27, 2013

Computational Geometry and Mesh Generation11 references3 citations

TL;DR

This paper proves that the 14 graphs obtained by $ abla\mathrm{Y}$ moves on $K_7$ are the only minor-minimal intrinsically knotted (MMIK) graphs with 21 edges. Using induction on decreasing graph order and leveraging the fact that $ abla\mathrm{Y}$ moves preserve intrinsic knotting, the authors show no other 21-edge graph can be MMIK, completing the classification of MMIK graphs by edge count.

ABSTRACT

We show that the 14 graphs obtained by $ abla\mathrm{Y}$ moves on K_7 constitute a complete list of the minor minimal intrinsically knotted graphs on 21 edges. We also present evidence in support of a conjecture that the 20 graph Heawood family, obtained by a combination of $ abla\mathrm{Y}$ and $\mathrm{Y} abla$ moves on K_7, is the list of graphs of size 21 that are minor minimal with respect to the property not 2--apex.

Motivation & Objective

• To complete the classification of minor-minimal intrinsically knotted (MMIK) graphs by edge count, specifically for graphs of size 21.
• To prove that the 14 graphs formed by $ abla\mathrm{Y}$ moves on $K_7$ (the KS graphs) are the only MMIK graphs with 21 edges.
• To investigate whether the Heawood family—generated by $ abla\mathrm{Y}$ and $\mathrm{Y}\nabla$ moves on $K_7$—contains all minor-minimal non-2-apex (MMN2A) graphs of size 21.
• To examine the preservation of the non-2-apex property under $\mathrm{Y}\nabla$ moves, particularly in the context of size-21 graphs.

Proposed method

• Use induction on decreasing graph order, starting from 14-vertex graphs down to 11–13 vertices, to analyze possible MMIK graphs of size 21.
• Leverage the fact that $ abla\mathrm{Y}$ moves preserve intrinsic knotting, a property attributed to Sachs, to maintain IK status through graph transformations.
• Apply degree sequence analysis and vertex deletion arguments: for a graph $G$ of size 21, if $G - a, b$ is planar for any two vertices $a, b$, then $G$ is not N2A, contradicting the assumption of being MMIK or MMN2A.
• Use known classifications of small non-planar graphs (e.g., (7,10), (7,11), (8,12), (8,13)) to eliminate candidate graphs via case analysis.
• Apply the $YT$ move (a form of $ abla\mathrm{Y}$) to reduce graphs with degree-three vertices to smaller graphs, relying on Proposition 1.4 to preserve the N2A property.
• Use symmetry and adjacency constraints in degree-4 and degree-5 graphs to eliminate configurations where $G - a, b$ would be planar, thus contradicting N2A or MMIK status.

Experimental results

Research questions

• RQ1 Are the 14 KS graphs the only minor-minimal intrinsically knotted graphs with 21 edges?
• RQ2 Is the Heawood family—the set of 20 graphs obtained from $K_7$ via $ abla\mathrm{Y}$ and $\mathrm{Y}\nabla$ moves—the complete list of minor-minimal non-2-apex (MMN2A) graphs of size 21?
• RQ3 Does the $\mathrm{Y}\nabla$ move preserve the non-2-apex property in graphs of size 21?
• RQ4 What is the smallest graph that is non-2-apex but becomes 2-apex after a $\mathrm{Y}\nabla$ move?

Key findings

• The 14 graphs obtained by $ abla\mathrm{Y}$ moves on $K_7$ are the only minor-minimal intrinsically knotted (MMIK) graphs with 21 edges.
• All connected 21-edge intrinsically knotted graphs are minor-minimal, as size-20 graphs are known to be non-IK.
• For graphs of order 11, 12, or 13 with 21 edges, any non-2-apex graph must be part of the Heawood family.
• Graphs of order 14 or more that are minor-minimal non-2-apex must be Heawood graphs, as they must have minimum degree 3 and satisfy the same structural constraints.
• The degree sequence $\{4^8, 5^2\}$ with the two degree-5 vertices adjacent, and $G - a, b$ being a cubic graph on 8 vertices, leads to only one possibility: graph 20 of the Heawood family.
• If a 10-vertex graph of size 21 is non-2-apex and has minimum degree greater than 3, it must be graph 20 of the Heawood family, confirming its uniqueness in this class.