# L11n153

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n153 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(2)^2 t(1)^2-2 t(2) t(1)^2-4 t(2)^2 t(1)+7 t(2) t(1)-4 t(1)-2 t(2)+2}{t(1) t(2)}$ (db) Jones polynomial $\frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)-2 a^9 z-a^9 z^{-1} +a^7 z^5+3 a^7 z^3+4 a^7 z+2 a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-2 a^3 z^3-4 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^6-4 a^{12} z^4+4 a^{12} z^2+2 a^{11} z^7-7 a^{11} z^5+6 a^{11} z^3-a^{11} z+2 a^{10} z^8-6 a^{10} z^6+6 a^{10} z^4-6 a^{10} z^2+2 a^{10}+a^9 z^9-a^9 z^7-5 a^9 z^3+3 a^9 z-a^9 z^{-1} +4 a^8 z^8-13 a^8 z^6+20 a^8 z^4-18 a^8 z^2+5 a^8+a^7 z^9-a^7 z^7+3 a^7 z^5-7 a^7 z^3+7 a^7 z-2 a^7 z^{-1} +2 a^6 z^8-5 a^6 z^6+10 a^6 z^4-7 a^6 z^2+3 a^6+2 a^5 z^7-4 a^5 z^5+7 a^5 z^3-2 a^5 z+a^4 z^6+a^4 z^2-a^4+3 a^3 z^3-5 a^3 z+a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        21-1
-6       41 3
-8      43  -1
-10     43   1
-12    34    1
-14   34     -1
-16  13      2
-18 13       -2
-20 1        1
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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