# L11a309

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a309 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2+5 t(1) t(2)^2-t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{25}{q^{5/2}}+\frac{21}{q^{7/2}}-\frac{16}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^5 z^5-2 a^5 z^3-2 a^5 z-a^5 z^{-1} +a^3 z^7+3 a^3 z^5+4 a^3 z^3+4 a^3 z+3 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +3 a z^3-2 z^3 a^{-1} -a z-z a^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+4 a^7 z^3+8 a^6 z^8-18 a^6 z^6+13 a^6 z^4-6 a^6 z^2+a^6+8 a^5 z^9-14 a^5 z^7+7 a^5 z^5-7 a^5 z^3+4 a^5 z-a^5 z^{-1} +3 a^4 z^{10}+11 a^4 z^8-37 a^4 z^6+31 a^4 z^4-13 a^4 z^2+3 a^4+16 a^3 z^9-31 a^3 z^7+20 a^3 z^5+z^5 a^{-3} -10 a^3 z^3-z^3 a^{-3} +7 a^3 z-3 a^3 z^{-1} +3 a^2 z^{10}+13 a^2 z^8-36 a^2 z^6+4 z^6 a^{-2} +28 a^2 z^4-5 z^4 a^{-2} -8 a^2 z^2+z^2 a^{-2} +3 a^2+8 a z^9-5 a z^7+8 z^7 a^{-1} -8 a z^5-12 z^5 a^{-1} +9 a z^3+7 z^3 a^{-1} +a z-2 z a^{-1} -2 a z^{-1} +10 z^8-14 z^6+7 z^4-z^2$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        93  -6
0       136   7
-2      1311    -2
-4     1211     1
-6    913      4
-8   712       -5
-10  39        6
-12 17         -6
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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