# L11a201

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a201 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(1)^2 t(2)^4-5 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-5 t(2)+2}{t(1) t(2)^2}$ (db) Jones polynomial $q^{7/2}-2 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{11}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^3 z^7-a z^7+a^5 z^5-4 a^3 z^5-5 a z^5+z^5 a^{-1} +3 a^5 z^3-4 a^3 z^3-9 a z^3+4 z^3 a^{-1} +2 a^5 z-a^3 z-8 a z+4 z a^{-1} +a^5 z^{-1} -2 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $a^9 z^3+3 a^8 z^4+6 a^7 z^5-5 a^7 z^3+2 a^7 z+8 a^6 z^6-10 a^6 z^4+2 a^6 z^2+9 a^5 z^7-18 a^5 z^5+10 a^5 z^3-5 a^5 z+a^5 z^{-1} +7 a^4 z^8-14 a^4 z^6+a^4 z^4+3 a^4 z^2-a^4+4 a^3 z^9-6 a^3 z^7-9 a^3 z^5+9 a^3 z^3-a^3 z+a^2 z^{10}+6 a^2 z^8+z^8 a^{-2} -34 a^2 z^6-6 z^6 a^{-2} +41 a^2 z^4+12 z^4 a^{-2} -16 a^2 z^2-9 z^2 a^{-2} +3 a^2+2 a^{-2} +6 a z^9+2 z^9 a^{-1} -26 a z^7-11 z^7 a^{-1} +35 a z^5+20 z^5 a^{-1} -21 a z^3-14 z^3 a^{-1} +10 a z+4 z a^{-1} -2 a z^{-1} - a^{-1} z^{-1} +z^{10}-18 z^6+39 z^4-26 z^2+5$ (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 \
-6-5-4-3-2-1012345χ
8           1-1
6          1 1
4         31 -2
2        41  3
0       53   -2
-2      64    2
-4     66     0
-6    55      0
-8   36       3
-10  35        -2
-12  3         3
-1413          -2
-161           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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