# L11a299

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a299 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(2 t(2) t(1)-t(1)-t(2)+2) \left(t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $q^{23/2}-3 q^{21/2}+6 q^{19/2}-10 q^{17/2}+12 q^{15/2}-13 q^{13/2}+13 q^{11/2}-11 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $z^5 a^{-9} +3 z^3 a^{-9} +2 z a^{-9} -z^7 a^{-7} -4 z^5 a^{-7} -5 z^3 a^{-7} -3 z a^{-7} -z^7 a^{-5} -4 z^5 a^{-5} -4 z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -3 z^3 a^{-13} +z a^{-13} +5 z^6 a^{-12} -5 z^4 a^{-12} +2 z^2 a^{-12} +6 z^7 a^{-11} -7 z^5 a^{-11} +3 z^3 a^{-11} +z a^{-11} +5 z^8 a^{-10} -5 z^6 a^{-10} +z^4 a^{-10} +3 z^9 a^{-9} -z^7 a^{-9} -3 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +4 z^8 a^{-8} -12 z^6 a^{-8} +8 z^4 a^{-8} -5 z^2 a^{-8} +5 z^9 a^{-7} -13 z^7 a^{-7} +10 z^5 a^{-7} -6 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +z^8 a^{-6} -10 z^6 a^{-6} +9 z^4 a^{-6} -2 z^2 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -2 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1}$ (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          2 2
20         41 -3
18        62  4
16       64   -2
14      76    1
12     77     0
10    46      -2
8   37       4
6  24        -2
4 14         3
2 1          -1
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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