# L10a113

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a113 at Knotilus!

 Planar diagram presentation X12,1,13,2 X8493 X18,14,19,13 X20,8,11,7 X6,20,7,19 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17 Gauss code {1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, {9, -1, 3, -7, 6, -8, 10, -3, 5, -4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(t(1)-1)\left(t(1)^{2}-t(1)+1\right)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}}$ (db) Jones polynomial ${\displaystyle 10q^{9/2}-11q^{7/2}+11q^{5/2}-{\frac {1}{q^{5/2}}}-12q^{3/2}+{\frac {3}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+8{\sqrt {q}}-{\frac {6}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+2az-7za^{-1}+8za^{-3}-3za^{-5}+az^{-1}-a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -2z^{9}a^{-1}-2z^{9}a^{-3}-10z^{8}a^{-2}-7z^{8}a^{-4}-3z^{8}-az^{7}+z^{7}a^{-1}-9z^{7}a^{-3}-11z^{7}a^{-5}+32z^{6}a^{-2}+10z^{6}a^{-4}-10z^{6}a^{-6}+12z^{6}+4az^{5}+18z^{5}a^{-1}+44z^{5}a^{-3}+24z^{5}a^{-5}-6z^{5}a^{-7}-23z^{4}a^{-2}+10z^{4}a^{-4}+16z^{4}a^{-6}-3z^{4}a^{-8}-14z^{4}-6az^{3}-30z^{3}a^{-1}-44z^{3}a^{-3}-16z^{3}a^{-5}+3z^{3}a^{-7}-z^{3}a^{-9}-11z^{2}a^{-4}-8z^{2}a^{-6}+3z^{2}+4az+14za^{-1}+16za^{-3}+6za^{-5}+1-az^{-1}-a^{-1}z^{-1}}$ (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 \
-4-3-2-10123456χ
16          11
14         2 -2
12        41 3
10       62  -4
8      54   1
6     66    0
4    65     1
2   48      4
0  24       -2
-2 14        3
-4 2         -2
-61          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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