# L11a522

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a522 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2 w^3-2 u^2 v^2 w^2+u^2 v^2 w-2 u^2 v w^3+4 u^2 v w^2-3 u^2 v w+u^2 v+u^2 w^3-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-4 u v^2 w+u v^2+3 u v w^3-8 u v w^2+8 u v w-3 u v-u w^3+4 u w^2-3 u w+u-v^2 w^2+2 v^2 w-v^2-v w^3+3 v w^2-4 v w+2 v-w^2+2 w-1}{u v w^{3/2}}$ (db) Jones polynomial $q^6-4 q^5+10 q^4-15 q^3+22 q^2-24 q+25-21 q^{-1} +16 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-9 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^8+5 z^6+10 z^4+8 z^2+ z^{-2} +3$ (db) Kauffman polynomial $3 z^{10} a^{-2} +3 z^{10}+9 a z^9+17 z^9 a^{-1} +8 z^9 a^{-3} +11 a^2 z^8+13 z^8 a^{-2} +8 z^8 a^{-4} +16 z^8+8 a^3 z^7-10 a z^7-36 z^7 a^{-1} -14 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-52 z^6 a^{-2} -19 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-11 a^3 z^5-2 a z^5+21 z^5 a^{-1} +3 z^5 a^{-3} -8 z^5 a^{-5} -5 a^4 z^4+15 a^2 z^4+62 z^4 a^{-2} +16 z^4 a^{-4} -2 z^4 a^{-6} +64 z^4-a^5 z^3+4 a^3 z^3+9 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +3 z^3 a^{-5} +a^4 z^2-7 a^2 z^2-37 z^2 a^{-2} -12 z^2 a^{-4} +z^2 a^{-6} -32 z^2-a^3 z-3 a z-8 z a^{-1} -6 z a^{-3} +a^2+12 a^{-2} +6 a^{-4} +8+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         71 6
7        94  -5
5       136   7
3      1210    -2
1     1312     1
-1    913      4
-3   712       -5
-5  310        7
-7 16         -5
-9 3          3
-111           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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