Invertible Knot

Author

Eric W. Weisstein
May 26, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/KnotTheory/InvertibleKnot.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/InvertibleKnot.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

KnotData[#,"Noninvertible"]&/@KnotData[All]

...wrong...
only 2 of the 9-crossing knots are noninvertible

According to Sloane's A042403, 33 of the 10-crossing knots are noninvertible

Pers. comm, Charles Livingston, May 16, 2008, the following are nonreversible:

Length[noninv=Sort[{91,92,93,94,95,98,102,106,107,110,117,119,147,148,149,150,151,153,79,81,88,109,115,118}]]

{10,#}&/@noninv

KnotData["Noninvertible"]

...wrong...

Accoridng to Charles Livingston,

"The data on KnotInfo includes symmetry type: chiral, reversible,
negative amphicheiral, positive amphicheiral, and fully amphicheiral.

Those of the type chiral, negative amphicheiral, and positive
amphicheiral, are not reversible. (The term reversible is used for
what was once called invertible "string orientation reversed", since
the inverse of a knot in the concordance group is not represented by
the reverse). For 10 or fewer crossings, this tabulation was done by
Hartley in the 70s. Now, KnotScape can determine reversibility."

Here is the complete list of (24) nonreversible prime knots on 10 crossings from
Charles L:

{{10, 79}, {10, 81}, {10, 88}, {10, 91}, {10, 92}, {10, 93}, {10, 
  94}, {10, 95}, {10, 98}, {10, 102}, {10, 106}, {10, 107}, {10, 
  109}, {10, 110}, {10, 115}, {10, 117}, {10, 118}, {10, 119}, {10, 
  147}, {10, 148}, {10, 149}, {10, 150}, {10, 151}, {10, 153}}

Show[KnotData[#,"KnotDiagram"],PlotLabel->TraditionalForm[KnotData[#,"AlexanderBriggsNotation"]]]&/@{{8,17},{9,32},{9,33}}

Show[KnotData[#],PlotLabel->TraditionalForm[KnotData[#,"AlexanderBriggsNotation"]]]&/@{{8,17},{9,32},{9,33}}