# L11a331

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a331 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^3 t(2)^5-3 t(1)^3 t(2)^4+3 t(1)^2 t(2)^4+3 t(1)^3 t(2)^3-6 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-6 t(1) t(2)^2+3 t(2)^2+3 t(1) t(2)-3 t(2)+1}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^7 z^7+5 a^7 z^5+8 a^7 z^3+6 a^7 z+2 a^7 z^{-1} -a^5 z^9-7 a^5 z^7-18 a^5 z^5-22 a^5 z^3-13 a^5 z-3 a^5 z^{-1} +a^3 z^7+5 a^3 z^5+7 a^3 z^3+3 a^3 z+a^3 z^{-1}$ (db) Kauffman polynomial $-z^3 a^{13}-3 z^4 a^{12}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}-8 z^6 a^{10}+10 z^4 a^{10}-2 z^2 a^{10}-9 z^7 a^9+17 z^5 a^9-7 z^3 a^9+3 z a^9-8 z^8 a^8+18 z^6 a^8-6 z^4 a^8+z^2 a^8-6 z^9 a^7+17 z^7 a^7-12 z^5 a^7+10 z^3 a^7-8 z a^7+2 a^7 z^{-1} -2 z^{10} a^6-2 z^8 a^6+31 z^6 a^6-43 z^4 a^6+18 z^2 a^6-3 a^6-9 z^9 a^5+42 z^7 a^5-62 z^5 a^5+39 z^3 a^5-16 z a^5+3 a^5 z^{-1} -2 z^{10} a^4+5 z^8 a^4+10 z^6 a^4-32 z^4 a^4+20 z^2 a^4-3 a^4-3 z^9 a^3+16 z^7 a^3-27 z^5 a^3+16 z^3 a^3-3 z a^3+a^3 z^{-1} -z^8 a^2+5 z^6 a^2-8 z^4 a^2+5 z^2 a^2-a^2$ (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 \
-8-7-6-5-4-3-2-10123χ
2           1-1
0          2 2
-2         31 -2
-4        52  3
-6       64   -2
-8      64    2
-10     66     0
-12    56      -1
-14   36       3
-16  35        -2
-18  3         3
-2013          -2
-221           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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