# L11a540

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a540 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(x-1) \left(u v w x^2-2 u v w x+2 u v x-u v-u w x^2+3 u w x-2 u x+2 u-2 v w x^2+2 v w x-3 v x+v+w x^2-2 w x+2 x-1\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db) Jones polynomial $q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z^{-3} +a^9 z+2 a^9 z^{-1} -3 a^7 z^3-3 a^7 z^{-3} -9 a^7 z-9 a^7 z^{-1} +3 a^5 z^5+11 a^5 z^3+3 a^5 z^{-3} +17 a^5 z+12 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-8 a^3 z^3-a^3 z^{-3} -10 a^3 z-5 a^3 z^{-1} +a z^5+2 a z^3+a z$ (db) Kauffman polynomial $-z^5 a^{11}+3 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -2 z^6 a^{10}+3 z^4 a^{10}-a^{10}-3 z^7 a^9+3 z^5 a^9-z^3 a^9+3 z a^9-3 a^9 z^{-1} +a^9 z^{-3} -3 z^8 a^8-2 z^6 a^8+12 z^4 a^8-16 z^2 a^8-3 a^8 z^{-2} +11 a^8-3 z^9 a^7-z^7 a^7+9 z^5 a^7-20 z^3 a^7+20 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -z^{10} a^6-9 z^8 a^6+19 z^6 a^6-3 z^4 a^6-28 z^2 a^6-6 a^6 z^{-2} +24 a^6-7 z^9 a^5+4 z^7 a^5+23 z^5 a^5-39 z^3 a^5+29 z a^5-14 a^5 z^{-1} +3 a^5 z^{-3} -z^{10} a^4-12 z^8 a^4+32 z^6 a^4-16 z^4 a^4-12 z^2 a^4-3 a^4 z^{-2} +13 a^4-4 z^9 a^3-2 z^7 a^3+28 z^5 a^3-31 z^3 a^3+18 z a^3-6 a^3 z^{-1} +a^3 z^{-3} -6 z^8 a^2+12 z^6 a^2-2 z^4 a^2-z^2 a^2-4 z^7 a+10 z^5 a-8 z^3 a+3 z a-z^6+2 z^4-z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          3 3
0         51 -4
-2        93  6
-4       98   -1
-6      106    4
-8     79     2
-10    910      -1
-12   411       7
-14  25        -3
-16  4         4
-1812          -1
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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