# L11a232

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a232 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-4 t(1)^2 t(2)^3+11 t(1) t(2)^3-6 t(2)^3+7 t(1)^2 t(2)^2-17 t(1) t(2)^2+7 t(2)^2-6 t(1)^2 t(2)+11 t(1) t(2)-4 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $-10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{5}{q^{7/2}}-24 q^{5/2}+\frac{11}{q^{5/2}}+27 q^{3/2}-\frac{18}{q^{3/2}}-q^{13/2}+4 q^{11/2}-28 \sqrt{q}+\frac{24}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} -a^3 z^3+4 z^3 a^{-3} +5 z a^{-3} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +3 a z^3-6 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +12 z^8 a^{-4} +a^4 z^6-17 z^6 a^{-4} -a^4 z^4+11 z^4 a^{-4} -3 z^2 a^{-4} +9 z^9 a^{-3} +5 a^3 z^7-9 a^3 z^5-24 z^5 a^{-3} +4 a^3 z^3+27 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +3 z^{10} a^{-2} +10 a^2 z^8+21 z^8 a^{-2} -21 a^2 z^6-53 z^6 a^{-2} +12 a^2 z^4+38 z^4 a^{-2} -a^2 z^2-10 z^2 a^{-2} + a^{-2} +9 a z^9+18 z^9 a^{-1} -9 a z^7-23 z^7 a^{-1} -12 a z^5-13 z^5 a^{-1} +13 a z^3+25 z^3 a^{-1} -4 a z-12 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^{10}+19 z^8-54 z^6+36 z^4-7 z^2$ (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 \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         71 6
8        103  -7
6       147   7
4      1411    -3
2     1413     1
0    1115      4
-2   713       -6
-4  411        7
-6 17         -6
-8 4          4
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{14}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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