# L11a343

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a343 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -3 a^7 z^3-7 a^7 z-5 a^7 z^{-1} +3 a^5 z^5+10 a^5 z^3+14 a^5 z+8 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-8 a^3 z^3-9 a^3 z-4 a^3 z^{-1} +a z^5+2 a z^3+a z$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+5 a^{10} z^2-2 a^{10}+4 a^9 z^7-3 a^9 z^5-2 a^9 z^3+a^9 z+a^9 z^{-1} +4 a^8 z^8+3 a^8 z^6-18 a^8 z^4+22 a^8 z^2-9 a^8+3 a^7 z^9+6 a^7 z^7-17 a^7 z^5+16 a^7 z^3-9 a^7 z+5 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-13 a^6 z^6-12 a^6 z^4+29 a^6 z^2-14 a^6+7 a^5 z^9-27 a^5 z^5+33 a^5 z^3-21 a^5 z+8 a^5 z^{-1} +a^4 z^{10}+12 a^4 z^8-26 a^4 z^6+5 a^4 z^4+13 a^4 z^2-8 a^4+4 a^3 z^9+2 a^3 z^7-24 a^3 z^5+25 a^3 z^3-14 a^3 z+4 a^3 z^{-1} +6 a^2 z^8-12 a^2 z^6+3 a^2 z^4+2 a^2 z^2+4 a z^7-10 a z^5+8 a z^3-2 a z+z^6-2 z^4+z^2$ (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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          3 3
0         51 -4
-2        93  6
-4       97   -2
-6      117    4
-8     89     1
-10    811      -3
-12   48       4
-14  28        -6
-16 14         3
-18 2          -2
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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