# L11a291

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a291 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-2 t(1)^2 t(2)^2+2 t(1) t(2)^2-2 t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-q^{11/2}+5 q^{9/2}-10 q^{7/2}+15 q^{5/2}-21 q^{3/2}+22 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-z^5 a^{-3} -2 a^3 z^3-z^3 a^{-3} +2 z a^{-3} +a z^7+z^7 a^{-1} +3 a z^5+2 z^5 a^{-1} +3 a z^3-2 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-4 z^{10} a^{-2} -4 z^{10}-11 a z^9-19 z^9 a^{-1} -8 z^9 a^{-3} -14 a^2 z^8-2 z^8 a^{-2} -5 z^8 a^{-4} -11 z^8-12 a^3 z^7+16 a z^7+54 z^7 a^{-1} +25 z^7 a^{-3} -z^7 a^{-5} -8 a^4 z^6+23 a^2 z^6+35 z^6 a^{-2} +15 z^6 a^{-4} +51 z^6-4 a^5 z^5+14 a^3 z^5+3 a z^5-37 z^5 a^{-1} -20 z^5 a^{-3} +2 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-10 a^2 z^4-34 z^4 a^{-2} -11 z^4 a^{-4} -40 z^4+2 a^5 z^3-4 a^3 z^3-9 a z^3-z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a^4 z^2-a^2 z^2+5 z^2 a^{-2} +5 z^2+4 a z+8 z a^{-1} +4 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-10123456χ
12           11
10          4 -4
8         61 5
6        94  -5
4       126   6
2      109    -1
0     1312     1
-2    912      3
-4   511       -6
-6  39        6
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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