# L11a226

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a226 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{5 u^2 v^3-8 u^2 v^2+5 u^2 v-u^2+3 u v^4-11 u v^3+17 u v^2-11 u v+3 u-v^4+5 v^3-8 v^2+5 v}{u v^2}$ (db) Jones polynomial $-\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{10}{q^{23/2}}-\frac{17}{q^{21/2}}+\frac{23}{q^{19/2}}-\frac{27}{q^{17/2}}+\frac{27}{q^{15/2}}-\frac{24}{q^{13/2}}+\frac{17}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z a^{13}-a^{13} z^{-1} +4 z^3 a^{11}+6 z a^{11}+2 a^{11} z^{-1} -3 z^5 a^9-5 z^3 a^9-z a^9-4 z^5 a^7-10 z^3 a^7-7 z a^7-a^7 z^{-1} -z^5 a^5-z^3 a^5$ (db) Kauffman polynomial $a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3-a^{15} z+8 a^{14} z^8-17 a^{14} z^6+13 a^{14} z^4-6 a^{14} z^2+2 a^{14}+8 a^{13} z^9-10 a^{13} z^7-3 a^{13} z^5+4 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^{10}+16 a^{12} z^8-51 a^{12} z^6+49 a^{12} z^4-25 a^{12} z^2+5 a^{12}+18 a^{11} z^9-30 a^{11} z^7+13 a^{11} z^5-10 a^{11} z^3+9 a^{11} z-2 a^{11} z^{-1} +3 a^{10} z^{10}+21 a^{10} z^8-56 a^{10} z^6+46 a^{10} z^4-17 a^{10} z^2+3 a^{10}+10 a^9 z^9-6 a^9 z^7-9 a^9 z^5+6 a^9 z^3+13 a^8 z^8-19 a^8 z^6+9 a^8 z^4+a^8 z^2-a^8+10 a^7 z^7-16 a^7 z^5+14 a^7 z^3-7 a^7 z+a^7 z^{-1} +4 a^6 z^6-3 a^6 z^4+a^5 z^5-a^5 z^3$ (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          41-3
-8         7  7
-10        104  -6
-12       147   7
-14      1310    -3
-16     1414     0
-18    1014      4
-20   713       -6
-22  310        7
-24 17         -6
-26 3          3
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-7$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-6$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=-5$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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