# L11n78

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n78 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^9-2 z a^9-2 a^9 z^{-1} +z^5 a^7+3 z^3 a^7+5 z a^7+4 a^7 z^{-1} +z^5 a^5+2 z^3 a^5+z a^5-a^5 z^{-1} -2 z^3 a^3-4 z a^3-a^3 z^{-1}$ (db) Kauffman polynomial $-z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-3 z^3 a^{11}-z a^{11}-2 z^8 a^{10}+4 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}-z^9 a^9+3 z^5 a^9+4 z^3 a^9-6 z a^9+2 a^9 z^{-1} -4 z^8 a^8+11 z^6 a^8-14 z^4 a^8+15 z^2 a^8-6 a^8-z^9 a^7+z^5 a^7+3 z^3 a^7-9 z a^7+4 a^7 z^{-1} -2 z^8 a^6+5 z^6 a^6-12 z^4 a^6+12 z^2 a^6-5 a^6-2 z^7 a^5+4 z^5 a^5-7 z^3 a^5+z a^5+a^5 z^{-1} -z^6 a^4-z^2 a^4+a^4-3 z^3 a^3+5 z a^3-a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         22
-4        32-1
-6       3  3
-8      43  -1
-10     53   2
-12    24    2
-14   45     -1
-16  12      1
-18 14       -3
-20 1        1
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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