# L11n116

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n116 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^5+2 t(1) t(2)^4-4 t(2)^4-7 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-7 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{3}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5-3 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^3-5 a^3 z-2 a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^6-2 a^{12} z^4+a^{12} z^2+4 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3+5 a^{10} z^8-11 a^{10} z^6+4 a^{10} z^4+a^{10} z^2-a^{10}+2 a^9 z^9+7 a^9 z^7-28 a^9 z^5+19 a^9 z^3-4 a^9 z+a^9 z^{-1} +11 a^8 z^8-22 a^8 z^6+7 a^8 z^4+2 a^8 z^2-3 a^8+2 a^7 z^9+10 a^7 z^7-31 a^7 z^5+27 a^7 z^3-12 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-7 a^6 z^6+a^6 z^4+5 a^6 z^2-3 a^6+7 a^5 z^7-14 a^5 z^5+22 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +3 a^4 z^6+3 a^4 z^2-2 a^4+6 a^3 z^3-7 a^3 z+2 a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         33
-4        41-3
-6       72 5
-8      64  -2
-10     87   1
-12    77    0
-14   47     -3
-16  37      4
-18 14       -3
-20 3        3
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$

### Computer Talk

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