# L11a524

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a524 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(v-1) \left(u^2 v w^3-3 u^2 v w^2+2 u^2 v w-u^2 w^3+2 u^2 w^2-u^2 w-u v w^3+4 u v w^2-5 u v w+u v+u w^3-5 u w^2+4 u w-u-v w^2+2 v w-v+2 w^2-3 w+1\right)}{u v w^{3/2}}$ (db) Jones polynomial $q^6-4 q^5- q^{-5} +10 q^4+5 q^{-4} -16 q^3-11 q^{-3} +24 q^2+18 q^{-2} -26 q-24 q^{-1} +28$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -a^2 z^6-2 z^6 a^{-2} -2 a^2 z^4-6 z^4 a^{-2} -5 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-2 a^{-2} +z^8+4 z^6+5 z^4+z^2+ z^{-2}$ (db) Kauffman polynomial $z^6 a^{-6} -2 z^4 a^{-6} +z^2 a^{-6} +4 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} +4 z^3 a^{-5} +8 z^8 a^{-4} +5 a^4 z^6-18 z^6 a^{-4} -4 a^4 z^4+16 z^4 a^{-4} -10 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +8 z^9 a^{-3} +11 a^3 z^7-11 z^7 a^{-3} -14 a^3 z^5-z^5 a^{-3} +4 a^3 z^3+6 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +3 z^{10} a^{-2} +14 a^2 z^8+16 z^8 a^{-2} -19 a^2 z^6-52 z^6 a^{-2} +6 a^2 z^4+50 z^4 a^{-2} -22 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+6 a^{-2} +10 a z^9+18 z^9 a^{-1} -2 a z^7-28 z^7 a^{-1} -20 a z^5+2 z^5 a^{-1} +13 a z^3+11 z^3 a^{-1} -5 z a^{-1} +2 a^{-1} z^{-1} +3 z^{10}+22 z^8-57 z^6+42 z^4-11 z^2- z^{-2} +2$ (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        104  -6
5       146   8
3      1311    -2
1     1513     2
-1    1014      4
-3   814       -6
-5  411        7
-7 17         -6
-9 4          4
-111           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{15}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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