Invertible knot

It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.{{citation

| last = Trotter | first = H. F.

| journal = Topology

| mr = 0158395

| pages = 275–280

| title = Non-invertible knots exist

| volume = 2

| year = 1963

| issue = 4

| doi=10.1016/0040-9383(63)90011-9| doi-access = free

}}. It is now known almost all knots are non-invertible.{{citation|title=Knot Theory and Its Applications|first=Kunio|last=Murasugi|publisher=Springer|year=2007|isbn=9780817647186|page=45|url=https://books.google.com/books?id=H2rdoTKwQNMC&pg=PA45}}.

File:Knot-trefoil-dir-128.png. Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow's direction reversed.]]

All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible.{{MathWorld|title=Invertible Knot|urlname=InvertibleKnot}} Accessed: May 5, 2013. The problem can be translated into algebraic terms,{{citation

| last = Kuperberg | first = Greg

| arxiv = q-alg/9712048

| doi = 10.1142/S021821659600014X

| issue = 2

| journal = Journal of Knot Theory and Its Ramifications

| mr = 1395778

| pages = 173–181

| title = Detecting knot invertibility

| volume = 5

| year = 1996| s2cid = 15295630

}}. but unfortunately there is no known algorithm to solve this algebraic problem.

If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot.{{citation

| last1 = Clark | first1 = W. Edwin

| last2 = Elhamdadi | first2 = Mohamed

| last3 = Saito | first3 = Masahico

| last4 = Yeatman | first4 = Timothy

| arxiv = 1312.3307

| title = Quandle colorings of knots and applications

| journal = Journal of Knot Theory and Its Ramifications

| year = 2013| volume = 23

| issue = 6

| doi = 10.1142/S0218216514500357

| pmid = 26491208

| pmc = 4610146

| bibcode = 2013arXiv1312.3307C}}.

A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot. All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible.{{citation

| last = Morimoto | first = Kanji

| doi = 10.1090/S0002-9939-1995-1317043-4

| issue = 11

| journal = Proceedings of the American Mathematical Society

| jstor = 2161103

| mr = 1317043

| pages = 3527–3532

| title = There are knots whose tunnel numbers go down under connected sum

| volume = 123

| year = 1995| doi-access = free

}}. See in particular Lemma 5.

Image:8 17 Knot.svg

The simplest example of a non-invertible knot is the knot 8₁₇ (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.

• Chiral knot

{{reflist}}

• Jablan, Slavik & Sazdanovic, Radmila. [http://math.ict.edu.rs:8080/webMathematica/LinkSL/ni01.htm Basic graph theory: Non-invertible knot and links] {{Webarchive|url=https://web.archive.org/web/20110118073721/http://math.ict.edu.rs:8080/webMathematica/LinkSL/ni01.htm |date=2011-01-18 }}, LinKnot.
• [http://nrich.maths.org/content/id/5861/VideoPlayer.html Explanation with a video], Nrich.Maths.org.

{{Knot theory|state=collapsed}}