More intrinsically knotted graphs with 22 edges and the restoring method

Hyoungjun Kim, Thomas Mattman, Seung Sang Oh

Research output: Contribution to journalArticle


A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman, showed that intrinsically knotted graphs have at least 21 edges. Recently, Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that K7 and the 13 graphs obtained from K7 by Y moves are the only intrinsically knotted graphs with 21 edges. Also Kim, Lee, Lee, Mattman and Oh showed that there are exactly three triangle-free intrinsically knotted graphs with 22 edges having at least two vertices of degree 5. Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. In this paper, we show that there are exactly five triangle-free intrinsically knotted graphs with 22 edges having exactly one degree 5 vertex. These are Cousin 29 of the K3,3,1,1 family, Cousins 97 and 99 of the E9 + e family and two others that were previously unknown.

Original languageEnglish
JournalJournal of Knot Theory and its Ramifications
Publication statusAccepted/In press - 2018 Jan 1



  • Graph embedding
  • intrinsically knotted

ASJC Scopus subject areas

  • Algebra and Number Theory

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