# L11n318

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n318 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^4-3 u v w^3+5 u v w^2-3 u v w+u w^3-2 u w^2+2 u w-2 v^2 w^3+2 v^2 w^2-v^2 w+3 v w^3-5 v w^2+3 v w-v-w^3+2 w^2-w}{\sqrt{u} v w^2}$ (db) Jones polynomial $- q^{-11} +4 q^{-10} -8 q^{-9} +11 q^{-8} -14 q^{-7} +15 q^{-6} -12 q^{-5} +11 q^{-4} -5 q^{-3} +3 q^{-2}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-z^2 a^{10}-a^{10} z^{-2} -a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8 z^{-2} +8 a^8-2 z^6 a^6-9 z^4 a^6-17 z^2 a^6-5 a^6 z^{-2} -15 a^6+3 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} +8 a^4$ (db) Kauffman polynomial $a^{13} z^5-a^{13} z^3+4 a^{12} z^6-6 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-13 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +6 a^{10} z^8-8 a^{10} z^6+2 a^{10} z^4-3 a^{10} z^2-a^{10} z^{-2} +2 a^{10}+2 a^9 z^9+9 a^9 z^7-28 a^9 z^5+27 a^9 z^3-13 a^9 z+5 a^9 z^{-1} +11 a^8 z^8-27 a^8 z^6+35 a^8 z^4-27 a^8 z^2-4 a^8 z^{-2} +13 a^8+2 a^7 z^9+5 a^7 z^7-20 a^7 z^5+33 a^7 z^3-27 a^7 z+9 a^7 z^{-1} +5 a^6 z^8-15 a^6 z^6+33 a^6 z^4-37 a^6 z^2-5 a^6 z^{-2} +20 a^6+3 a^5 z^7-6 a^5 z^5+13 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +6 a^4 z^4-14 a^4 z^2-2 a^4 z^{-2} +10 a^4$ (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 \
-9-8-7-6-5-4-3-2-10χ
-3         33
-5        42-2
-7       71 6
-9      65  -1
-11     96   3
-13    56    1
-15   69     -3
-17  36      3
-19 15       -4
-21 3        3
-231         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$

### Computer Talk

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