# L11a203

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a203 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^2 v^4-3 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^4+3 u v^3-3 u v^2+3 u v-2 u-2 v^3+3 v^2-3 v+2}{u v^2}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z^5 a^9-4 z^3 a^9-4 z a^9-a^9 z^{-1} +z^7 a^7+5 z^5 a^7+9 z^3 a^7+8 z a^7+2 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+3 z^3 a^5-z a^5-z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1}$ (db) Kauffman polynomial $-z^4 a^{14}+2 z^2 a^{14}-2 z^5 a^{13}+3 z^3 a^{13}-3 z^6 a^{12}+5 z^4 a^{12}-3 z^2 a^{12}-4 z^7 a^{11}+10 z^5 a^{11}-12 z^3 a^{11}+2 z a^{11}-4 z^8 a^{10}+12 z^6 a^{10}-18 z^4 a^{10}+9 z^2 a^{10}-2 a^{10}-3 z^9 a^9+9 z^7 a^9-13 z^5 a^9+9 z^3 a^9-4 z a^9+a^9 z^{-1} -z^{10} a^8-2 z^8 a^8+17 z^6 a^8-31 z^4 a^8+25 z^2 a^8-5 a^8-5 z^9 a^7+20 z^7 a^7-31 z^5 a^7+29 z^3 a^7-12 z a^7+2 a^7 z^{-1} -z^{10} a^6+10 z^6 a^6-14 z^4 a^6+10 z^2 a^6-3 a^6-2 z^9 a^5+6 z^7 a^5-z^5 a^5-3 z^3 a^5-z a^5-2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-z^2 a^4+a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
0           11
-2          1 -1
-4         41 3
-6        32  -1
-8       63   3
-10      54    -1
-12     55     0
-14    45      1
-16   35       -2
-18  14        3
-20 13         -2
-22 1          1
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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