71 knot

---

title: "71 knot" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public description: "Mathematical knot with crossing number 7" topic_path: "uncategorized" source: "https://en.wikipedia.org/wiki/71_knot" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical knot with crossing number 7 ::

::data[format=table title="Infobox knot theory"]

Field	Value
name	71 knot
image	Blue 7 1 Knot.png
arf invariant	0
braid length	7
braid number	2
bridge number	2
crosscap number	1
crossing number	7
genus	3
hyperbolic volume	0
stick number	9
unknotting number	3
conway_notation	[7]
ab_notation	71
dowker notation	8, 10, 12, 14, 2, 4, 6
last crossing	6
last order	3
next crossing	7
next order	2
alternating	alternating
class	torus
fibered	fibered
prime	prime
symmetry	reversible
::

| name= 71 knot | names= | image= Blue 7 1 Knot.png | caption= | arf invariant= 0 | braid length= 7 | braid number= 2 | bridge number= 2 | crosscap number= 1 | crossing number= 7 | genus= 3 | hyperbolic volume= 0 | linking number= | stick number= 9 | unknotting number= 3 | conway_notation= [7] | ab_notation= 71 | dowker notation= 8, 10, 12, 14, 2, 4, 6 | thistlethwaite= | last crossing= 6 | last order= 3 | next crossing= 7 | next order= 2 | alternating= alternating | class= torus | fibered= fibered | prime= prime | slice= | symmetry= reversible | tricolorable= In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil. This knot is used to construct the simplest counterexample to the conjecture that the unknotting number is additive under connected sum.

Properties

The 71 knot is invertible but not amphichiral. Its Alexander polynomial is

:\Delta(t) = t^3 - t^2 + t - 1 + t^{-1} - t^{-2} + t^{-3}, ,

its Conway polynomial is

:\nabla(z) = z^6 + 5z^4 + 6z^2 + 1, ,

and its Jones polynomial is

:V(q) = q^{-3} + q^{-5} - q^{-6} + q^{-7} - q^{-8} + q^{-9} - q^{-10}. ,

Example

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/88/7₁_knot.webm" caption="Assembling of 7₁ knot."] ::

References

References

1. (2025). "Unknotting number is not additive under connected sum".
2. Sloman, Leila. (2025-09-22). "A Simple Way To Measure Knots Has Come Unraveled".
3. {{Knot Atlas. 7_1

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::