# L11a403

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a403 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-5 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+5 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db) Jones polynomial $q^7-4 q^6+9 q^5-15 q^4+20 q^3-22 q^2+23 q-18+15 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-10 z^4 a^{-2} -2 a^2 z^2-9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +2 z^6+7 z^4+7 z^2-2 z^{-2} +1$ (db) Kauffman polynomial $2 z^{10} a^{-2} +2 z^{10}+5 a z^9+14 z^9 a^{-1} +9 z^9 a^{-3} +4 a^2 z^8+23 z^8 a^{-2} +15 z^8 a^{-4} +12 z^8+a^3 z^7-11 a z^7-28 z^7 a^{-1} -2 z^7 a^{-3} +14 z^7 a^{-5} -14 a^2 z^6-77 z^6 a^{-2} -25 z^6 a^{-4} +9 z^6 a^{-6} -57 z^6-3 a^3 z^5-2 a z^5-6 z^5 a^{-1} -30 z^5 a^{-3} -19 z^5 a^{-5} +4 z^5 a^{-7} +17 a^2 z^4+73 z^4 a^{-2} +15 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +67 z^4+3 a^3 z^3+13 a z^3+26 z^3 a^{-1} +27 z^3 a^{-3} +10 z^3 a^{-5} -z^3 a^{-7} -8 a^2 z^2-30 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -29 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        93  -6
7       116   5
5      1210    -2
3     1110     1
1    914      5
-1   69       -3
-3  310        7
-5 15         -4
-7 3          3
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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