# L11a41

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a41 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{15}{q^{9/2}}-q^{7/2}+\frac{21}{q^{7/2}}+5 q^{5/2}-\frac{26}{q^{5/2}}-12 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{8}{q^{11/2}}+18 \sqrt{q}-\frac{24}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -3 a^3 z^5-7 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +5 a z^3-z^3 a^{-1} +4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-7 a^7 z^5+6 a^7 z^3-3 a^7 z+a^7 z^{-1} +5 a^6 z^8-8 a^6 z^6+2 a^6 z^4+3 a^6 z^2-2 a^6+5 a^5 z^9-a^5 z^7-15 a^5 z^5+21 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-40 a^4 z^6+32 a^4 z^4-8 a^4 z^2+14 a^3 z^9-12 a^3 z^7-23 a^3 z^5+z^5 a^{-3} +33 a^3 z^3-16 a^3 z+3 a^3 z^{-1} +2 a^2 z^{10}+25 a^2 z^8-58 a^2 z^6+5 z^6 a^{-2} +39 a^2 z^4-3 z^4 a^{-2} -11 a^2 z^2+2 a^2+9 a z^9+4 a z^7+12 z^7 a^{-1} -32 a z^5-16 z^5 a^{-1} +24 a z^3+6 z^3 a^{-1} -9 a z-2 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +15 z^8-22 z^6+9 z^4-3 z^2$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          4 -4
4         81 7
2        104  -6
0       148   6
-2      1412    -2
-4     1212     0
-6    914      5
-8   612       -6
-10  29        7
-12 16         -5
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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