# L11a211

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a211 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-2 t(2)^3+6 t(1)^2 t(2)^2-11 t(1) t(2)^2+6 t(2)^2-2 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)-2 t(1)+2}{t(1) t(2)^2}$ (db) Jones polynomial $q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z^5 a^7+3 z^3 a^7+3 z a^7+a^7 z^{-1} -z^7 a^5-4 z^5 a^5-7 z^3 a^5-6 z a^5-a^5 z^{-1} -z^7 a^3-3 z^5 a^3-2 z^3 a^3+z^5 a+2 z^3 a$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+7 z^5 a^9-3 z^3 a^9+z a^9-6 z^8 a^8+8 z^6 a^8-5 z^4 a^8+2 z^2 a^8-5 z^9 a^7+5 z^7 a^7-4 z^5 a^7+5 z^3 a^7-4 z a^7+a^7 z^{-1} -2 z^{10} a^6-7 z^8 a^6+21 z^6 a^6-22 z^4 a^6+9 z^2 a^6-a^6-11 z^9 a^5+26 z^7 a^5-27 z^5 a^5+16 z^3 a^5-6 z a^5+a^5 z^{-1} -2 z^{10} a^4-8 z^8 a^4+30 z^6 a^4-28 z^4 a^4+8 z^2 a^4-6 z^9 a^3+12 z^7 a^3-5 z^5 a^3+z^3 a^3-7 z^8 a^2+19 z^6 a^2-14 z^4 a^2+3 z^2 a^2-4 z^7 a+10 z^5 a-5 z^3 a-z^6+2 z^4$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          3 3
0         51 -4
-2        83  5
-4       106   -4
-6      107    3
-8     910     1
-10    810      -2
-12   49       5
-14  38        -5
-16 15         4
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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