# L11n306

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n306 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^4-u v^2 w^3+u v^2 w^2-u v^2 w+u v w-u v+u-v^2 w^4+v w^4-v w^3+w^3-w^2+w-1}{\sqrt{u} v w^2}$ (db) Jones polynomial $- q^{-10} +2 q^{-9} -3 q^{-8} +4 q^{-7} -4 q^{-6} +5 q^{-5} -3 q^{-4} +4 q^{-3} - q^{-2} + q^{-1}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+a^8 z^6+6 a^8 z^4+12 a^8 z^2+4 a^8 z^{-2} +11 a^8-a^6 z^8-7 a^6 z^6-18 a^6 z^4-24 a^6 z^2-5 a^6 z^{-2} -18 a^6+a^4 z^6+6 a^4 z^4+12 a^4 z^2+2 a^4 z^{-2} +9 a^4$ (db) Kauffman polynomial $z a^{13}+2 z^2 a^{12}-a^{12}+z^5 a^{11}-z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} +3 z^6 a^{10}-10 z^4 a^{10}+6 z^2 a^{10}-a^{10} z^{-2} +5 z^7 a^9-23 z^5 a^9+31 z^3 a^9-19 z a^9+5 a^9 z^{-1} +4 z^8 a^8-19 z^6 a^8+27 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+z^9 a^7+z^7 a^7-25 z^5 a^7+50 z^3 a^7-35 z a^7+9 a^7 z^{-1} +5 z^8 a^6-29 z^6 a^6+55 z^4 a^6-46 z^2 a^6-5 a^6 z^{-2} +22 a^6+z^9 a^5-4 z^7 a^5-z^5 a^5+18 z^3 a^5-19 z a^5+5 a^5 z^{-1} +z^8 a^4-7 z^6 a^4+18 z^4 a^4-21 z^2 a^4-2 a^4 z^{-2} +11 a^4$ (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 \
-7-6-5-4-3-2-1012χ
-1         11
-3          0
-5       41 3
-7      12  1
-9     42   2
-11   122    1
-13   33     0
-15 122      1
-17 23       -1
-19 1        1
-211         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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