# L11n447

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n447 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(w-1) (x-1) \left(-u v+u x^2-u x+u+v x^2-v x+v-x^2\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}$ (db) Jones polynomial $-\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{7}{q^{11/2}}+2 \sqrt{q}-\frac{6}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^9 z^{-1} +a^9 z^{-3} -4 z a^7-7 a^7 z^{-1} -3 a^7 z^{-3} +5 z^3 a^5+13 z a^5+12 a^5 z^{-1} +3 a^5 z^{-3} -2 z^5 a^3-7 z^3 a^3-11 z a^3-7 a^3 z^{-1} -a^3 z^{-3} +2 z^3 a+2 z a+a z^{-1}$ (db) Kauffman polynomial $a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+6 a^8 z^2+3 a^8 z^{-2} -9 a^8+a^7 z^9+5 a^7 z^7-34 a^7 z^5+55 a^7 z^3-3 a^7 z^{-3} -38 a^7 z+14 a^7 z^{-1} +8 a^6 z^8-22 a^6 z^6+a^6 z^4+27 a^6 z^2+6 a^6 z^{-2} -21 a^6+a^5 z^9+15 a^5 z^7-65 a^5 z^5+83 a^5 z^3-3 a^5 z^{-3} -54 a^5 z+18 a^5 z^{-1} +6 a^4 z^8-8 a^4 z^6-18 a^4 z^4+33 a^4 z^2+3 a^4 z^{-2} -18 a^4+11 a^3 z^7-35 a^3 z^5+45 a^3 z^3-a^3 z^{-3} -31 a^3 z+11 a^3 z^{-1} +7 a^2 z^6-14 a^2 z^4+15 a^2 z^2-6 a^2+a z^5+7 a z^3-5 a z+2 a z^{-1} +3 z^2-1$ (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 \
-8-7-6-5-4-3-2-101χ
2         2-2
0        4 4
-2       64 -2
-4      531 3
-6     46   2
-8    85    3
-10   510     5
-12  12      -1
-14 15       4
-16 1        -1
-181         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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