# L11n109

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n109 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^3-u v^2-u v+u+v^3-v^2-v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +2 z a^{11}+2 a^{11} z^{-1} +z a^9-a^9 z^{-1} -z^5 a^7-4 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-3 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $-z^8 a^{14}+7 z^6 a^{14}-15 z^4 a^{14}+11 z^2 a^{14}-3 a^{14}-z^9 a^{13}+7 z^7 a^{13}-15 z^5 a^{13}+12 z^3 a^{13}-4 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+15 z^6 a^{12}-34 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-z^9 a^{11}+8 z^7 a^{11}-20 z^5 a^{11}+21 z^3 a^{11}-10 z a^{11}+2 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-18 z^4 a^{10}+14 z^2 a^{10}-4 a^{10}+2 z^3 a^9-4 z a^9+a^9 z^{-1} -z^6 a^8+4 z^4 a^8-3 z^2 a^8-z^7 a^7+4 z^5 a^7-3 z^3 a^7-z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+4 z^3 a^5-3 z a^5+a^5 z^{-1}$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          110
-8         1  1
-10       111  1
-12      131   1
-14       1    1
-16    122     -1
-18   1        1
-20   11       0
-22 11         0
-24            0
-261           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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