# L11n81

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n81 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^2+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{8}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{8}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^9-a^9 z^{-1} -z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+5 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-5 z a^3-2 a^3 z^{-1}$ (db) Kauffman polynomial $-3 z^2 a^{12}+a^{12}-z^5 a^{11}-3 z^3 a^{11}+z a^{11}-3 z^6 a^{10}+3 z^4 a^{10}-2 z^2 a^{10}-4 z^7 a^9+8 z^5 a^9-6 z^3 a^9+2 z a^9+a^9 z^{-1} -3 z^8 a^8+5 z^6 a^8-z^4 a^8+6 z^2 a^8-3 a^8-z^9 a^7-4 z^7 a^7+21 z^5 a^7-19 z^3 a^7+4 z a^7+a^7 z^{-1} -5 z^8 a^6+16 z^6 a^6-11 z^4 a^6+2 z^2 a^6-z^9 a^5-z^7 a^5+17 z^5 a^5-25 z^3 a^5+10 z a^5-2 a^5 z^{-1} -2 z^8 a^4+8 z^6 a^4-7 z^4 a^4-3 z^2 a^4+3 a^4-z^7 a^3+5 z^5 a^3-9 z^3 a^3+7 z a^3-2 a^3 z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
0         11
-2        1 -1
-4       41 3
-6      33  0
-8     52   3
-10    33    0
-12   45     -1
-14  23      1
-16 24       -2
-18 2        2
-202         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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