# L11a486

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a486 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1)^2 \left(t(3)^3+2 t(2) t(3)^2-2 t(3)^2-2 t(2) t(3)+2 t(3)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}}$ (db) Jones polynomial $q^9-4 q^8+10 q^7-17 q^6+23 q^5-25 q^4+27 q^3-22 q^2+17 q-9+4 q^{-1} - q^{-2}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-8} + a^{-8} -3 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +2 z^6 a^{-4} +5 z^4 a^{-4} +4 z^2 a^{-4} -2 a^{-4} z^{-2} - a^{-4} +z^6 a^{-2} +z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^4-z^2$ (db) Kauffman polynomial $2 z^{10} a^{-4} +2 z^{10} a^{-6} +8 z^9 a^{-3} +15 z^9 a^{-5} +7 z^9 a^{-7} +11 z^8 a^{-2} +25 z^8 a^{-4} +22 z^8 a^{-6} +8 z^8 a^{-8} +8 z^7 a^{-1} +z^7 a^{-3} -14 z^7 a^{-5} -3 z^7 a^{-7} +4 z^7 a^{-9} -16 z^6 a^{-2} -67 z^6 a^{-4} -65 z^6 a^{-6} -17 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-10 z^5 a^{-1} -22 z^5 a^{-3} -23 z^5 a^{-5} -20 z^5 a^{-7} -8 z^5 a^{-9} +12 z^4 a^{-2} +63 z^4 a^{-4} +62 z^4 a^{-6} +14 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+5 z^3 a^{-1} +19 z^3 a^{-3} +27 z^3 a^{-5} +20 z^3 a^{-7} +6 z^3 a^{-9} -4 z^2 a^{-2} -27 z^2 a^{-4} -30 z^2 a^{-6} -8 z^2 a^{-8} +z^2 a^{-10} +2 z^2-z a^{-1} -z a^{-3} -5 z a^{-5} -7 z a^{-7} -2 z a^{-9} - a^{-2} +2 a^{-4} +5 a^{-6} +3 a^{-8} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (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 \
-3-2-1012345678χ
19           11
17          3 -3
15         71 6
13        103  -7
11       137   6
9      1412    -2
7     1311     2
5    914      5
3   813       -5
1  311        8
-1 16         -5
-3 3          3
-51           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=4$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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