# L11a214

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a214 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^2 t(2)^4-3 t(1) t(2)^4-4 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+6 t(1)^2 t(2)^2-13 t(1) t(2)^2+6 t(2)^2-4 t(1)^2 t(2)+9 t(1) t(2)-4 t(2)-3 t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+3 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-4 a^3 z-a z-z a^{-1} +z a^{-3} +a^5 z^{-1} -2 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-6 a^3 z^9-12 a z^9-6 z^9 a^{-1} -7 a^4 z^8-12 a^2 z^8-7 z^8 a^{-2} -12 z^8-6 a^5 z^7+5 a^3 z^7+22 a z^7+7 z^7 a^{-1} -4 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+34 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +40 z^6-a^7 z^5+10 a^5 z^5-a^3 z^5-14 a z^5+7 z^5 a^{-1} +9 z^5 a^{-3} +4 a^6 z^4-2 a^4 z^4-35 a^2 z^4-10 z^4 a^{-2} +2 z^4 a^{-4} -41 z^4+2 a^7 z^3-9 a^5 z^3+9 a z^3-7 z^3 a^{-1} -5 z^3 a^{-3} -a^6 z^2-2 a^4 z^2+16 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} +22 z^2-a^7 z+5 a^5 z-7 a z+z a^{-3} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 a z^{-1} + a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         61 -5
4        83  5
2       116   -5
0      118    3
-2     1112     1
-4    810      -2
-6   511       6
-8  38        -5
-10  5         5
-1213          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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