# L11a385

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a385 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 t(1) t(3)^3+2 t(1) t(2) t(3)^3-4 t(2) t(3)^3+2 t(3)^3+6 t(1) t(3)^2-4 t(1) t(2) t(3)^2+7 t(2) t(3)^2-4 t(3)^2-7 t(1) t(3)+4 t(1) t(2) t(3)-6 t(2) t(3)+4 t(3)+4 t(1)-2 t(1) t(2)+2 t(2)-2}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^8+4 q^7-10 q^6+14 q^5-18 q^4+21 q^3+ q^{-3} -19 q^2-2 q^{-2} +17 q+7 q^{-1} -10$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2-3 a^{-2} +6 a^{-4} -2 a^{-6} -3+a^2 z^{-2} -2 a^{-2} z^{-2} +3 a^{-4} z^{-2} - a^{-6} z^{-2} - z^{-2}$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +8 z^8 a^{-2} +15 z^8 a^{-4} +10 z^8 a^{-6} +3 z^8+2 a z^7+6 z^7 a^{-1} +z^7 a^{-3} +6 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-15 z^6 a^{-2} -31 z^6 a^{-4} -17 z^6 a^{-6} +4 z^6 a^{-8} -4 z^6-4 a z^5-23 z^5 a^{-1} -33 z^5 a^{-3} -31 z^5 a^{-5} -16 z^5 a^{-7} +z^5 a^{-9} -4 a^2 z^4+6 z^4 a^{-2} +18 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} -2 z^4+23 z^3 a^{-1} +50 z^3 a^{-3} +39 z^3 a^{-5} +11 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2-3 z^2 a^{-2} -4 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2+4 a z-10 z a^{-1} -34 z a^{-3} -27 z a^{-5} -7 z a^{-7} -4 a^2+4 a^{-2} +5 a^{-4} + a^{-6} -3-2 a z^{-1} +2 a^{-1} z^{-1} +10 a^{-3} z^{-1} +8 a^{-5} z^{-1} +2 a^{-7} z^{-1} +a^2 z^{-2} -2 a^{-2} z^{-2} -3 a^{-4} z^{-2} - a^{-6} z^{-2} + z^{-2}$ (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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        73  4
9       117   -4
7      107    3
5     911     2
3    810      -2
1   512       7
-1  25        -3
-3  5         5
-512          -1
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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