# L11a76

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a76 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 t(2)^3+6 t(1) t(2)^2-9 t(2)^2-9 t(1) t(2)+6 t(2)+4 t(1)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{9}{q^{17/2}}-\frac{7}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +3 z a^{11}+a^{11} z^{-1} -2 z^3 a^9+z a^9+2 a^9 z^{-1} -4 z^3 a^7-4 z a^7-2 a^7 z^{-1} -3 z^3 a^5-2 z a^5-z^3 a^3$ (db) Kauffman polynomial $a^{14} z^8-6 a^{14} z^6+13 a^{14} z^4-12 a^{14} z^2+4 a^{14}+2 a^{13} z^9-10 a^{13} z^7+15 a^{13} z^5-6 a^{13} z^3-a^{13} z^{-1} +a^{12} z^{10}+a^{12} z^8-23 a^{12} z^6+48 a^{12} z^4-34 a^{12} z^2+9 a^{12}+6 a^{11} z^9-23 a^{11} z^7+20 a^{11} z^5+3 a^{11} z^3-3 a^{11} z-a^{11} z^{-1} +a^{10} z^{10}+7 a^{10} z^8-38 a^{10} z^6+46 a^{10} z^4-19 a^{10} z^2+4 a^{10}+4 a^9 z^9-4 a^9 z^7-19 a^9 z^5+26 a^9 z^3-12 a^9 z+2 a^9 z^{-1} +7 a^8 z^8-13 a^8 z^6-2 a^8 z^4+5 a^8 z^2-2 a^8+9 a^7 z^7-18 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +8 a^6 z^6-10 a^6 z^4+2 a^6 z^2+6 a^5 z^5-5 a^5 z^3+2 a^5 z+3 a^4 z^4+a^3 z^3$ (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
-2           11
-4          31-2
-6         3  3
-8        53  -2
-10       63   3
-12      66    0
-14     65     1
-16    36      3
-18   46       -2
-20  13        2
-22 14         -3
-24 1          1
-261           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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