View source for L10a140 link

{{Short description|Link of three loops with ten crossings}}
{{Infobox knot theory
| name=              L10a140
| practical name=    
| image=    Brunnian-L10a140.svg
| caption=           
| arf invariant=     
| braid length=      10
| braid number=      3
| bridge number=     
| crossing number=   10
| hyperbolic volume= 12.27627758
| linking number=    
| stick number=      
| unknotting number= 
| conway_notation=[.3:30]
| ab_notation=       
| dowker notation=   
| thistlethwaite=    L10a140
| other=             
 | alternating=      alternating
 | fibered=          
 | slice=            
 | symmetry=         
 | tricolorable=     
| last knot=         L10a139
| next knot=         L10a141
}}

In the [[Knot theory|mathematical theory of knots]], '''L10a140''' is the name in the [http://katlas.math.toronto.edu/wiki/The_Thistlethwaite_Link_Table Thistlethwaite link table] of a [[Link (knot theory)|link]] of three loops, which has ten crossings between the loops when presented in its simplest visual form.<ref>{{Knot Atlas|L10a140}}</ref> It is of interest because it is presumably the simplest link which possesses the Brunnian property — a link of connected components that, when one component is removed, becomes entirely unconnected<ref>[[Biography:Colin Adams (mathematician)|Adams, Colin C.]] (1994). ''The Knot Book'', {{page needed|date=April 2013}}. American Mathematical Society. {{isbn|9780716723936}}.</ref> — other than the six-crossing [[Borromean rings]].<ref>[[Biography:Dror Bar-Natan|Bar-Natan, Dror]] (2010-08-16). "[http://drorbn.net/AcademicPensieve/2010-08/nb/All%20Brunnians,%20Maybe.pdf All Brunnians, Maybe]", ''Academic Pensieve''.</ref>

In other words, no two loops are directly linked with each other, but all three are collectively interlinked, so removing any loop frees the other two. In the image in the infobox at right, the red loop is not interlinked with either the blue or the yellow loops, and if the red loop is removed, then the blue and yellow loops can also be disentangled from each other without cutting either one.

According to work by Slavik V. Jablan, the L10a140 link can be seen as the second in an infinite series of Brunnian links beginning with the Borromean rings. So if the blue and yellow loops have only one twist along each side, the resulting configuration is the Borromean rings; if the blue and yellow loops have three twists along each side, the resulting configuration is the L10a140 link; if the blue and yellow loops have five twists along each side, the resulting configuration is a three-loop link with 14 overall crossings, etc. etc.<ref>Jablan, Slavik V., ''Are Borromean Links So Rare?'', Forma '''14''' (1999), 269&ndash;277. Online at the electronic journal ''[http://www.mi.sanu.ac.rs/vismath/bor/bor4.htm Vismath]''. L10a140 is depicted in the middle figure of the top image.</ref>

==Invariants==
The [http://katlas.org/wiki/The_Multivariable_Alexander_Polynomial multivariable Alexander polynomial] for the L10a140 link is

: <math>\Delta(u,v,w)=\frac{(u-1)(v-1)(w-1)(vw+1)^2}{vw\sqrt{uvw}}, \, </math>

the Conway polynomial is

: <math>\nabla (z)=4z^4+4z^6+z^8, \, </math>

the [[Jones polynomial]] factors nicely as

: <math>\begin{align}V(t) &= -t^5+3t^4-5t^3+8t^2-9t+12-9t^{-1}+8t^{-2}-5t^{-3}+3t^{-4}-t^{-5} \\[8pt]
& = -\frac{1}{t^5}\left(t^5-2 t^4+t^3-2t^2+t-1\right)
  \left(t^5-t^4+2 t^3-t^2+2t-1\right) \\[8pt]
& = w(t) w(1/t), \,
\end{align}
</math>
where <math>w(t) = t^5-2 t^4+t^3-2 t^2+t-1.</math> (Notice that <math>w(t)</math> is essentially the Jones polynomial for the [[Whitehead link]].)

The [[HOMFLY polynomial]] is

: <math>P(\alpha,z)=z^{-2} \alpha^{-2}-4z^2 \alpha^{-2}-4z^4 \alpha^{-2}-z^6 \alpha^{-2}-2z^{-2}+8z^2+12z^4+6z^6+z^8+z^{-2} \alpha^2-4z^2 \alpha^2-4z^4 \alpha^2-z^6 \alpha^2, \, </math>

and the [[Kauffman polynomial]] is

: <math>
\begin{align}
F(a,z) & = 1 + 2z^{-2} + a^{-2}z^{-2} + a^{2}z^{-2} - 2a^{-1}z^{-1} - az^{-1} - 20z^2 + 2a^{-4}z^2 \\[6pt]
& {} - 8a^{-2}z^2 - 8a^2 z^2 + 2a^4 z^4 + 2a^4 z^2 - 2a^{-5} z^3 + 4a^{-3} z^3 + 6a^{-1} z^3 \\[6pt]
& {} + 6az^3 +4a^3 z^3 - 2a^5 z^3 +42z^4 - 7a^{-4} z^4 +14a^{-2} z^{4} \\[6pt]
& {} + 14a^2 z^4 -7a^4 z^4 + a^{-5} z^{5} - 9a^{-3}z^5 - 2a^{-1} z^5 - 2az^5 - 9a^3 z^5 \\[6pt]
& {} + a^5 z^5 - 28z^6 +3a^{-4} z^6 -11a^{-2} z^6 -11a^2 z^6 + 3a^4 z^6 + 4a^{-3} z^7 \\[6pt]
& {} - 2a^{-1} z^7 - 2az^7 + 4a^3 z^7 + 8z^8 + 4a^{-2} z^8 + 4a^2 z^8 + 2a^{-1} z^9 + 2az^9.
\end{align}
</math>

==Pseudo-symmetric visual variants==
David Swart,<ref>Dror Bar-Natan (2010-08-14). "[http://drorbn.net/AcademicPensieve/2010-08/one/A_Link_from_David_Swart.pdf A Link from David Swart]", ''[Academic Pensieve]''.</ref> and independently Rick Mabry and Laura McCormick,<ref>{{cite journal |last=Swart|first=David |title=It is what it is |journal=[[Math Horizons]] |volume=18|issue=4 |date=April 2011}}</ref> discovered alternative 12-crossing visual representations of the L10a140 link. In these depictions, the link no longer has strictly alternating crossings (as it does in its simplest 10-crossing form), but there is greater superficial symmetry.

So the leftmost image below shows a 12-crossing link (distinct from both the Borromean rings and the L10a140 link) with six-fold rotational symmetry. The center image shows a similar-looking depiction of the L10a140 link (but without true rotational symmetry). Similarly, the rightmost image shows a depiction of the L10a140 link with superficial fourfold symmetry.

<gallery>
Image:Brunnian-3-not-Borromean.svg|Fully symmetrical 12-crossing Brunnian link (L12a1882)
File:Marilyns Cross.gif|L10a140 in pseudo 6-symmetric form
File:Marilyns Cross The Cross.gif|L10a140 in pseudo 4-symmetric form
</gallery>

== References ==
{{Reflist}}

== External links ==
*[https://www.flickr.com/photos/dmswart/4878510547/ "It Is What It Is"], ''Flickr.com''.

{{Knot theory|state=collapsed}}

[[Category:Hyperbolic knots and links]]

{{Sourceattribution|L10a140 link}}