# L11a12

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a12 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{21}{q^{5/2}}-\frac{25}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z-3 a^7 z^3-4 a^7 z-a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+6 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-2 a^3 z+a z^5+a z^3-a z-a z^{-1}$ (db) Kauffman polynomial $a^{11} z^5-a^{11} z^3+4 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+7 a^9 z^7-9 a^9 z^5+5 a^9 z^3-2 a^9 z+8 a^8 z^8-5 a^8 z^6-7 a^8 z^4+9 a^8 z^2-2 a^8+6 a^7 z^9+5 a^7 z^7-25 a^7 z^5+24 a^7 z^3-8 a^7 z+a^7 z^{-1} +2 a^6 z^{10}+17 a^6 z^8-36 a^6 z^6+15 a^6 z^4+8 a^6 z^2-5 a^6+13 a^5 z^9-10 a^5 z^7-26 a^5 z^5+31 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+18 a^4 z^8-47 a^4 z^6+27 a^4 z^4-3 a^4+7 a^3 z^9-3 a^3 z^7-21 a^3 z^5+18 a^3 z^3-4 a^3 z+9 a^2 z^8-19 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-10 a z^5+5 a z^3+a z-a z^{-1} +z^6-z^4$ (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          4 4
0         61 -5
-2        114  7
-4       128   -4
-6      139    4
-8     1112     1
-10    1013      -3
-12   511       6
-14  310        -7
-16 15         4
-18 3          -3
-201           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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