# L10n107

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n107 at Knotilus! L10n107 is the "Borromean chain mail" link - it contains two L6a4 configurations without any L2a1 configuration (i.e. no two loops are linked). Compare L10a169.
An indefinitely extended "Borromean chainmail" pattern made up of overlapping L10n107 links; no two circles are directly linked.
 "Borromean chain-mail" represented with circles Represented with minimally-overlapping same-size circles

 Planar diagram presentation X6172 X5,12,6,13 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18 Gauss code {1, -4, 3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, -3, -6, 8}, {4, -5, 10, 9, -8, 7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle 0}$ (db) Jones polynomial ${\displaystyle q^{9/2}+{\frac {1}{q^{9/2}}}-2q^{7/2}-{\frac {2}{q^{7/2}}}+q^{5/2}+{\frac {1}{q^{5/2}}}-2q^{3/2}-{\frac {2}{q^{3/2}}}-2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle -a^{3}z^{3}+z^{3}a^{-3}+a^{3}z^{-3}-a^{-3}z^{-3}-2a^{3}z+2za^{-3}+az^{5}-z^{5}a^{-1}+5az^{3}-5z^{3}a^{-1}-3az^{-3}+3a^{-1}z^{-3}+6az-6za^{-1}}$ (db) Kauffman polynomial ${\displaystyle -a^{2}z^{8}-z^{8}a^{-2}-2z^{8}-2a^{3}z^{7}-4az^{7}-4z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}+4a^{2}z^{6}+4z^{6}a^{-2}-z^{6}a^{-4}+10z^{6}+10a^{3}z^{5}+26az^{5}+26z^{5}a^{-1}+10z^{5}a^{-3}+4a^{4}z^{4}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}a^{-4}-4z^{4}-12a^{3}z^{3}-44az^{3}-44z^{3}a^{-1}-12z^{3}a^{-3}-2a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}-2z^{2}a^{-4}-12z^{2}+8a^{3}z+24az+24za^{-1}+8za^{-3}+1-3az^{-1}-3a^{-1}z^{-1}+3a^{2}z^{-2}+3a^{-2}z^{-2}+6z^{-2}-a^{3}z^{-3}-3az^{-3}-3a^{-1}z^{-3}-a^{-3}z^{-3}}$ (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
10          1-1
8         1 1
6       111 1
4       21  1
2     521   4
0    282    4
-2   125     4
-4  12       1
-6 111       1
-8 1         1
-101          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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