# L11n154

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n154 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^4-u^2 v^3+u^2 v^2-u^2 v-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u-v^3+v^2-v+1}{u v^2}$ (db) Jones polynomial $-\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{6}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 \left(-z^5\right)-4 a^9 z^3-5 a^9 z-2 a^9 z^{-1} +a^7 z^7+6 a^7 z^5+14 a^7 z^3+15 a^7 z+5 a^7 z^{-1} -2 a^5 z^5-9 a^5 z^3-11 a^5 z-3 a^5 z^{-1}$ (db) Kauffman polynomial $a^{14} z^4-2 a^{14} z^2+2 a^{13} z^5-3 a^{13} z^3+3 a^{12} z^6-6 a^{12} z^4+5 a^{12} z^2-a^{12}+3 a^{11} z^7-7 a^{11} z^5+9 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-4 a^{10} z^6+5 a^{10} z^4+a^9 z^9-2 a^9 z^7+5 a^9 z^5-6 a^9 z^3+5 a^9 z-2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+21 a^8 z^4-20 a^8 z^2+5 a^8+a^7 z^9-5 a^7 z^7+17 a^7 z^5-30 a^7 z^3+20 a^7 z-5 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+9 a^6 z^4-13 a^6 z^2+5 a^6+3 a^5 z^5-12 a^5 z^3+13 a^5 z-3 a^5 z^{-1}$ (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        110
-8       41 3
-10      32  -1
-12     43   1
-14    33    0
-16   34     -1
-18  13      2
-20 13       -2
-22 1        1
-241         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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