# L11n71

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n71 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^3-u v^2-2 u v+2 u+2 v^3-2 v^2-v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +2 z a^{11}+a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -z^5 a^7-5 z^3 a^7-5 z a^7-2 a^7 z^{-1} -z^5 a^5-4 z^3 a^5-2 z a^5$ (db) Kauffman polynomial $a^{14} z^8-7 a^{14} z^6+16 a^{14} z^4-14 a^{14} z^2+4 a^{14}+a^{13} z^9-6 a^{13} z^7+10 a^{13} z^5-5 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-20 a^{12} z^6+41 a^{12} z^4-33 a^{12} z^2+9 a^{12}+a^{11} z^9-5 a^{11} z^7+3 a^{11} z^5+5 a^{11} z^3-2 a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+22 a^{10} z^4-13 a^{10} z^2+4 a^{10}+2 a^9 z^7-14 a^9 z^5+24 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+8 a^8 z^2-2 a^8+a^7 z^7-6 a^7 z^5+10 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^6-4 a^6 z^4+2 a^6 z^2+a^5 z^5-4 a^5 z^3+2 a^5 z$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          110
-8        11  0
-10       111  1
-12      241   1
-14     1 1    2
-16    132     0
-18   21       1
-20   11       0
-22 12         -1
-24            0
-261           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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