# L11a227

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a227 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-3 u^2 v-2 u v^4+10 u v^3-15 u v^2+10 u v-2 u-3 v^3+8 v^2-5 v+1}{u v^2}$ (db) Jones polynomial $\frac{23}{q^{9/2}}-\frac{24}{q^{7/2}}+\frac{21}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{14}{q^{13/2}}-\frac{20}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -3 a^7 z^3-5 a^7 z-2 a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^5+a z^3-a z$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-6 z^7 a^9+9 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -8 z^8 a^8+12 z^6 a^8-11 z^4 a^8+4 z^2 a^8-6 z^9 a^7-z^7 a^7+19 z^5 a^7-26 z^3 a^7+12 z a^7-2 a^7 z^{-1} -2 z^{10} a^6-16 z^8 a^6+42 z^6 a^6-37 z^4 a^6+12 z^2 a^6-a^6-13 z^9 a^5+15 z^7 a^5+15 z^5 a^5-26 z^3 a^5+12 z a^5-2 a^5 z^{-1} -2 z^{10} a^4-17 z^8 a^4+48 z^6 a^4-34 z^4 a^4+8 z^2 a^4-7 z^9 a^3+5 z^7 a^3+16 z^5 a^3-14 z^3 a^3+5 z a^3-a^3 z^{-1} -9 z^8 a^2+20 z^6 a^2-11 z^4 a^2+z^2 a^2-5 z^7 a+10 z^5 a-4 z^3 a-z a-z^6+z^4$ (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          4 4
0         61 -5
-2        104  6
-4       127   -5
-6      129    3
-8     1112     1
-10    912      -3
-12   511       6
-14  39        -6
-16 16         5
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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