# L11n80

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n80 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z^5 a^9-4 z^3 a^9-6 z a^9-3 a^9 z^{-1} +z^7 a^7+6 z^5 a^7+15 z^3 a^7+18 z a^7+7 a^7 z^{-1} -2 z^5 a^5-9 z^3 a^5-12 z a^5-4 a^5 z^{-1}$ (db) Kauffman polynomial $a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-2 a^{13} z^3+3 a^{12} z^6-3 a^{12} z^4+a^{12} z^2+3 a^{11} z^7-3 a^{11} z^5+2 a^{11} z^3+2 a^{10} z^8-a^{10} z^6+a^{10} z^2+a^9 z^9-a^9 z^7+6 a^9 z^5-11 a^9 z^3+8 a^9 z-3 a^9 z^{-1} +3 a^8 z^8-8 a^8 z^6+15 a^8 z^4-19 a^8 z^2+7 a^8+a^7 z^9-4 a^7 z^7+14 a^7 z^5-27 a^7 z^3+22 a^7 z-7 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+11 a^6 z^4-17 a^6 z^2+7 a^6+3 a^5 z^5-12 a^5 z^3+14 a^5 z-4 a^5 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-4         22
-6        220
-8       4  4
-10      32  -1
-12     54   1
-14    33    0
-16   45     -1
-18  13      2
-20 14       -3
-22 1        1
-241         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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