Vol. 9, No. 4, 2016

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Editorial Login
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Author Index
Coming Soon
Other MSP Journals
This article is available for purchase or by subscription. See below.
Graphs on 21 edges that are not 2-apex

Jamison Barsotti and Thomas W. Mattman

Vol. 9 (2016), No. 4, 591–621

We show that the 20-graph Heawood family, obtained by a combination of Y and Y moves on K7, is precisely the set of graphs of at most 21 edges that are minor-minimal with respect to the property “not 2-apex”. As a corollary, this gives a new proof that the 14 graphs obtained by Y moves on K7 are the minor-minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven-graph Petersen family, obtained from K6, is the set of graphs of at most 17 edges that are minor-minimal with respect to the property “not apex”.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

spatial graphs, intrinsic knotting, apex graphs, forbidden minors
Mathematical Subject Classification 2010
Primary: 05C10
Secondary: 57M15, 57M25
Received: 12 January 2015
Revised: 23 June 2015
Accepted: 17 August 2015
Published: 6 July 2016

Communicated by Joel Foisy
Jamison Barsotti
Department of Mathematics
University of California
Santa Cruz, CA 95064
United States
Thomas W. Mattman
Department of Mathematics and Statistics
California State University
Chico, CA 95929-0525
United States