# L11n76

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n76 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^5-5 t(1) t(2)^4+t(2)^4+8 t(1) t(2)^3-6 t(2)^3-6 t(1) t(2)^2+8 t(2)^2+t(1) t(2)-5 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{9/2}-\frac{3}{q^{9/2}}-3 q^{7/2}+\frac{7}{q^{7/2}}+7 q^{5/2}-\frac{11}{q^{5/2}}-11 q^{3/2}+\frac{13}{q^{3/2}}+13 \sqrt{q}-\frac{15}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+2 z a^{-3} -2 a^3 z^{-1} + a^{-3} z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-6 z^3 a^{-1} +5 a z-6 z a^{-1} +3 a z^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a z^9-2 z^9 a^{-1} -7 a^2 z^8-4 z^8 a^{-2} -11 z^8-8 a^3 z^7-11 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -3 a^4 z^6+12 a^2 z^6+7 z^6 a^{-2} -z^6 a^{-4} +23 z^6+16 a^3 z^5+35 a z^5+27 z^5 a^{-1} +8 z^5 a^{-3} -3 a^4 z^4-13 a^2 z^4+2 z^4 a^{-2} +3 z^4 a^{-4} -11 z^4-6 a^5 z^3-21 a^3 z^3-33 a z^3-25 z^3 a^{-1} -7 z^3 a^{-3} +3 a^4 z^2+9 a^2 z^2-6 z^2 a^{-2} -3 z^2 a^{-4} +3 z^2+5 a^5 z+11 a^3 z+15 a z+12 z a^{-1} +3 z a^{-3} -2 a^2+2 a^{-2} + a^{-4} -a^5 z^{-1} -2 a^3 z^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-1}$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-4-3-2-1012345χ
10         1-1
8        2 2
6       51 -4
4      62  4
2     75   -2
0    86    2
-2   68     2
-4  57      -2
-6 26       4
-815        -4
-103         3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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