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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a140 at Knotilus!

Brunnian link. Presumably the simplest Brunnian link other than the Borromean rings.[1] The second in an infinite series of Brunnian links -- if the blue and yellow loops in the illustration below have only one twist along each side, the result is the Borromean rings; if the blue and yellow loops have three twists along each side, the result is this L10a140 link; if the blue and yellow loops have five twists along each side, the result is a three-loop link with 14 overall crossings, etc.[2]

In a visual form which makes it evident that it is a Brunnian link.

Link Presentations

[edit Notes on L10a140's Link Presentations]

Planar diagram presentation X6172 X2,16,3,15 X10,4,11,3 X14,6,15,5 X20,12,13,11 X12,14,5,13 X4,19,1,20 X8,17,9,18 X16,7,17,8 X18,9,19,10
Gauss code {1, -2, 3, -7}, {4, -1, 9, -8, 10, -3, 5, -6}, {6, -4, 2, -9, 8, -10, 7, -5}
A Braid Representative
A Morse Link Presentation L10a140 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(2) t(3)+1)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^5+3 q^4-5 q^3+8 q^2-9 q+12-9 q^{-1} +8 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6-z^6 a^{-2} -4 a^2 z^4-4 z^4 a^{-2} -4 a^2 z^2-4 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +z^8+6 z^6+12 z^4+8 z^2-2 z^{-2} (db)
Kauffman polynomial 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+4 a^3 z^7-2 a z^7-2 z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-11 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -28 z^6+a^5 z^5-9 a^3 z^5-2 a z^5-2 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+14 a^2 z^4+14 z^4 a^{-2} -7 z^4 a^{-4} +42 z^4-2 a^5 z^3+4 a^3 z^3+6 a z^3+6 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} +2 z^2 a^{-4} -20 z^2+1-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
11          1-1
9         2 2
7        31 -2
5       52  3
3      43   -1
1     85    3
-1    58     3
-3   34      -1
-5  25       3
-7 13        -2
-9 2         2
-111          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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