---
tag: ZSAMVFPWN6
kind: example
name: Conway knot
msc: [57K10]
instance_of: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Conway_knot
same_as:
  wikidata: Q94806552
  wikipedia: Conway knot
---

## Example

The *Conway knot* is the prime [[knot|QYEAZTYGPT]] with $11$ crossings listed as $11n34$ in the knot tables, named after John Horton Conway.
It is obtained from the Kinoshita–Terasaka knot, another $11$-crossing knot, by a mutation.
A mutation cuts out a ball that meets the knot in two arcs, rotates the ball by a half turn, and glues it back.
The result is again a knot, but in general not an equivalent one.

## Remarks

Mutation does not change many [[knot invariants|GZQ5ZE3X6G]].
The Conway knot and the Kinoshita–Terasaka knot have the same [[Jones polynomial|4SPC17FPXE]], the same HOMFLY polynomial, and both have [[Alexander polynomial|JKA25KDPQP]] equal to $1$, like the [[unknot|1TQ06R4Q66]].
They are still different knots: the Conway knot has genus $3$, while the Kinoshita–Terasaka knot has genus $2$, as shown by David Gabai.
Since its Alexander polynomial is $1$, the Conway knot is topologically [[slice|P68G28E4AD]], by a theorem of Michael Freedman.
The Kinoshita–Terasaka knot is smoothly slice.
Lisa Piccirillo proved in 2020 that the Conway knot is not smoothly slice, which answered a question that had been open for about fifty years.

## Explanations

### L1 human

The Conway knot is a knot with eleven crossings that fools many of the standard tests.
Its polynomial invariants agree with those of a close relative, and its Alexander polynomial even agrees with that of the unknot.
It became famous when a graduate student, Lisa Piccirillo, settled whether it bounds a smooth disk in four-dimensional space.