# L11a347

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a347 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2+2 t(1)^2-t(2)^2 t(1)-t(1)+2 t(2)^2-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+q^{3/2}+q^{23/2}-4 q^{21/2}+8 q^{19/2}-12 q^{17/2}+15 q^{15/2}-15 q^{13/2}+15 q^{11/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $-z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -5 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} +5 z^3 a^{-3} -11 z^3 a^{-5} +2 z^3 a^{-9} +8 z a^{-3} -14 z a^{-5} +6 z a^{-7} +4 a^{-3} z^{-1} -8 a^{-5} z^{-1} +5 a^{-7} z^{-1} - a^{-9} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-14} +4 z^5 a^{-13} -2 z^3 a^{-13} +8 z^6 a^{-12} -8 z^4 a^{-12} +2 z^2 a^{-12} +10 z^7 a^{-11} -13 z^5 a^{-11} +4 z^3 a^{-11} +z a^{-11} +8 z^8 a^{-10} -9 z^6 a^{-10} +3 z^2 a^{-10} -2 a^{-10} +4 z^9 a^{-9} -5 z^5 a^{-9} -5 z^3 a^{-9} +3 z a^{-9} + a^{-9} z^{-1} +z^{10} a^{-8} +6 z^8 a^{-8} -10 z^6 a^{-8} -11 z^4 a^{-8} +20 z^2 a^{-8} -9 a^{-8} +5 z^9 a^{-7} -10 z^7 a^{-7} +3 z^5 a^{-7} +3 z^3 a^{-7} -7 z a^{-7} +5 a^{-7} z^{-1} +z^{10} a^{-6} -z^8 a^{-6} +5 z^6 a^{-6} -25 z^4 a^{-6} +33 z^2 a^{-6} -14 a^{-6} +z^9 a^{-5} +z^7 a^{-5} -15 z^5 a^{-5} +27 z^3 a^{-5} -21 z a^{-5} +8 a^{-5} z^{-1} +z^8 a^{-4} -2 z^6 a^{-4} -5 z^4 a^{-4} +14 z^2 a^{-4} -8 a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +13 z^3 a^{-3} -12 z a^{-3} +4 a^{-3} z^{-1}$ (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 \
-2-10123456789χ
24           1-1
22          3 3
20         51 -4
18        73  4
16       85   -3
14      77    0
12     88     0
10    47      -3
8   38       5
6  24        -2
4 15         4
2            0
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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