# L11a47

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a47 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^2+1\right) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $3 q^{9/2}-\frac{4}{q^{9/2}}-6 q^{7/2}+\frac{8}{q^{7/2}}+10 q^{5/2}-\frac{13}{q^{5/2}}-15 q^{3/2}+\frac{16}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+17 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-z^5 a^{-3} -2 a^3 z^3-3 z^3 a^{-3} -2 z a^{-3} +a^3 z^{-1} - a^{-3} z^{-1} +a z^7+z^7 a^{-1} +3 a z^5+4 z^5 a^{-1} +a z^3+6 z^3 a^{-1} -3 a z+5 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4+z^7 a^{-5} +4 a^5 z^5-4 z^5 a^{-5} -2 a^5 z^3+4 z^3 a^{-5} +3 z^8 a^{-4} +8 a^4 z^6-12 z^6 a^{-4} -7 a^4 z^4+14 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} +4 z^9 a^{-3} +11 a^3 z^7-15 z^7 a^{-3} -15 a^3 z^5+20 z^5 a^{-3} +6 a^3 z^3-14 z^3 a^{-3} +a^3 z+5 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 z^{10} a^{-2} +10 a^2 z^8-14 a^2 z^6-14 z^6 a^{-2} +4 a^2 z^4+16 z^4 a^{-2} +a^2 z^2-5 z^2 a^{-2} +6 a z^9+10 z^9 a^{-1} -4 a z^7-31 z^7 a^{-1} -6 a z^5+37 z^5 a^{-1} -3 a z^3-29 z^3 a^{-1} +8 a z+12 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+7 z^8-24 z^6+14 z^4-z^2-1$ (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          2 -2
8         41 3
6        62  -4
4       94   5
2      86    -2
0     109     1
-2    810      2
-4   58       -3
-6  38        5
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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