# L11a210

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a210 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+6 t(1)^2 t(2)^2-11 t(1) t(2)^2+6 t(2)^2-4 t(1)^2 t(2)+9 t(1) t(2)-5 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2}$ (db) Jones polynomial $-q^{11/2}+4 q^{9/2}-9 q^{7/2}+14 q^{5/2}-20 q^{3/2}+22 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+3 a z^3+3 z^3 a^{-1} -2 z^3 a^{-3} -a^3 z+a z+2 z a^{-1} -z a^{-3} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $-3 z^{10} a^{-2} -3 z^{10}-10 a z^9-16 z^9 a^{-1} -6 z^9 a^{-3} -15 a^2 z^8-5 z^8 a^{-2} -4 z^8 a^{-4} -16 z^8-14 a^3 z^7+8 a z^7+40 z^7 a^{-1} +17 z^7 a^{-3} -z^7 a^{-5} -9 a^4 z^6+24 a^2 z^6+37 z^6 a^{-2} +13 z^6 a^{-4} +57 z^6-4 a^5 z^5+19 a^3 z^5+17 a z^5-20 z^5 a^{-1} -11 z^5 a^{-3} +3 z^5 a^{-5} -a^6 z^4+7 a^4 z^4-11 a^2 z^4-38 z^4 a^{-2} -13 z^4 a^{-4} -44 z^4+a^5 z^3-10 a^3 z^3-15 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -3 z^3 a^{-5} -2 a^4 z^2+2 a^2 z^2+10 z^2 a^{-2} +4 z^2 a^{-4} +10 z^2+2 a^3 z+3 a z+z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-5-4-3-2-10123456χ
12           11
10          3 -3
8         61 5
6        83  -5
4       126   6
2      119    -2
0     1211     1
-2    912      3
-4   611       -5
-6  39        6
-8 16         -5
-10 3          3
-121           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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