# L11n459

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n459 at Knotilus!

 Link L11n459. A graph, L11n459. A part of a knot and a part of a graph.

 Planar diagram presentation X6172 X12,3,13,4 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X4,11,1,12 Gauss code {1, -10, 2, -11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {11, -2, -5, 7, -9, 8, -7, 6}
A Braid Representative

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(wx-1)(uwx+u(-w)-ux+2u-2vwx+vw+vx-v)}{{\sqrt {u}}{\sqrt {v}}wx}}}$ (db) Jones polynomial ${\displaystyle -2q^{9/2}-q^{5/2}-{\frac {1}{q^{5/2}}}-2q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+q^{11/2}-{\frac {3}{\sqrt {q}}}}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle za^{-7}-z^{3}a^{-5}-a^{-5}z^{-3}-2za^{-5}-2a^{-5}z^{-1}+z^{3}a^{-3}+3a^{-3}z^{-3}+6za^{-3}+7a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-8za^{-1}+3az^{-1}-8a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-3z^{8}a^{-4}-z^{8}a^{-6}-z^{8}-az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}-4z^{7}a^{-5}-2z^{7}a^{-7}+17z^{6}a^{-2}+18z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}+4z^{6}+6az^{5}+3z^{5}a^{-1}+12z^{5}a^{-3}+25z^{5}a^{-5}+10z^{5}a^{-7}-18z^{4}a^{-2}-23z^{4}a^{-4}+4z^{4}a^{-8}+z^{4}-11az^{3}-19z^{3}a^{-1}-35z^{3}a^{-3}-39z^{3}a^{-5}-12z^{3}a^{-7}-10z^{2}a^{-2}-2z^{2}a^{-6}-2z^{2}a^{-8}-10z^{2}+10az+23za^{-1}+27za^{-3}+20za^{-5}+6za^{-7}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}}$ (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 \
-4-3-2-101234567χ
16           1-1
14          1 1
12        111 1
10       131  1
8      222   2
6     362    1
4    124     3
2   251      2
0  113       3
-2  2         2
-411          0
-61           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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