# L11n434

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n434 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v w-u^2 v-u^2 w+u^2+u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-2 u v w^3+3 u v w^2-3 u v w+2 u v+u w^3-u w^2+u w-u-v^2 w^3+v^2 w^2+v w^3-2 v w^2}{u v w^{3/2}}$ (db) Jones polynomial $q^6-3 q^5+6 q^4-7 q^3- q^{-3} +10 q^2+4 q^{-2} -9 q-6 q^{-1} +9$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^6 a^{-2} -4 z^4 a^{-2} +z^4 a^{-4} +2 z^4-a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +4 z^2-5 a^{-2} +2 a^{-4} +3-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} -9 z^5 a^{-5} +5 z^3 a^{-5} +4 z^8 a^{-4} -13 z^6 a^{-4} +13 z^4 a^{-4} -11 z^2 a^{-4} - a^{-4} z^{-2} +6 a^{-4} +2 z^9 a^{-3} -2 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+8 z^3 a^{-3} -a^3 z-6 z a^{-3} +2 a^{-3} z^{-1} +8 z^8 a^{-2} -30 z^6 a^{-2} +4 a^2 z^4+44 z^4 a^{-2} -4 a^2 z^2-33 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2+12 a^{-2} +2 z^9 a^{-1} +2 a z^7-3 z^7 a^{-1} -4 a z^5-2 z^5 a^{-1} +8 a z^3+10 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-16 z^6+32 z^4-24 z^2- z^{-2} +8$ (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 \
-3-2-10123456χ
13         11
11        2 -2
9       41 3
7      43  -1
5     63   3
3    45    1
1   55     0
-1  25      3
-3 24       -2
-5 3        3
-71         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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