L10a140

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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.

 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}

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-5-4-3-2-1012345χ
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
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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

Read me first: Modifying Knot Pages

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