# L11a333

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a333 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1) t(2)+1) \left(t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+4 a^3 z^5+5 a^3 z^3+3 a^3 z+2 a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -13 a z^3+5 z^3 a^{-1} -8 a z+3 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-9 a^2 z^8-7 z^8 a^{-2} -6 z^8-8 a^5 z^7+11 a^3 z^7+37 a z^7+14 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+18 a^4 z^6+32 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +27 z^6-a^7 z^5+13 a^5 z^5-5 a^3 z^5-37 a z^5-9 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-13 a^4 z^4-34 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -27 z^4+a^7 z^3-6 a^5 z^3+a^3 z^3+19 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +13 z^2+2 a^5 z-4 a^3 z-9 a z-3 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (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      128    4
-2     1112     1
-4    911      -2
-6   511       6
-8  49        -5
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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