# L11a507

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a507 at Knotilus!

 Planar diagram presentation X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22 Gauss code {1, -4, 3, -10, 9, -11}, {5, -1, 6, -3, 2, -7}, {10, -9, 8, -5, 7, -2, 4, -6, 11, -8}
A Braid Representative
A Morse Link Presentation

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(w-1)\left(u^{2}v^{2}w+u^{2}\left(-v^{2}\right)+u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w-2uv^{2}w+2uv^{2}-2uvw^{2}+5uvw-2uv+2uw^{2}-2uw+v^{2}w-v^{2}+vw^{2}-2vw+v-w^{2}+w\right)}{uvw^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{9}-4q^{8}+8q^{7}-13q^{6}+19q^{5}-20q^{4}+22q^{3}-18q^{2}-q^{-2}+14q+4q^{-1}-8}$ (db) Signature 2 (db) HOMFLY-PT polynomial ${\displaystyle z^{2}a^{-8}-2z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+z^{4}a^{-4}-2a^{-4}z^{-2}-2a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{4}-z^{2}}$ (db) Kauffman polynomial ${\displaystyle 2z^{10}a^{-4}+2z^{10}a^{-6}+6z^{9}a^{-3}+12z^{9}a^{-5}+6z^{9}a^{-7}+8z^{8}a^{-2}+12z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}a^{-8}+7z^{7}a^{-1}-3z^{7}a^{-3}-24z^{7}a^{-5}-10z^{7}a^{-7}+4z^{7}a^{-9}-10z^{6}a^{-2}-35z^{6}a^{-4}-41z^{6}a^{-6}-19z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-10z^{5}a^{-1}-6z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}-10z^{5}a^{-9}+2z^{4}a^{-2}+34z^{4}a^{-4}+44z^{4}a^{-6}+16z^{4}a^{-8}-2z^{4}a^{-10}-6z^{4}-az^{3}+3z^{3}a^{-1}+4z^{3}a^{-3}+5z^{3}a^{-7}+5z^{3}a^{-9}+z^{2}a^{-2}-10z^{2}a^{-4}-15z^{2}a^{-6}-6z^{2}a^{-8}+2z^{2}+2za^{-3}+2za^{-5}-2a^{-2}-3a^{-4}-2a^{-6}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}}$ (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 \
-3-2-1012345678χ
19           11
17          3 -3
15         51 4
13        83  -5
11       115   6
9      109    -1
7     1210     2
5    812      4
3   610       -4
1  39        6
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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