# L11n74

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n74 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u v^5+u v^4-u v^3+u v^2-u-v^7+v^5-v^4+v^3-v^2}{\sqrt{u} v^{7/2}}$ (db) Jones polynomial $3 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{q^{3/2}}+q^{17/2}-q^{15/2}+2 q^{13/2}-3 q^{11/2}-\sqrt{q}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +2 z a^{-7} +2 z^3 a^{-5} +5 z a^{-5} +3 a^{-5} z^{-1} -z^7 a^{-3} -8 z^5 a^{-3} -20 z^3 a^{-3} -19 z a^{-3} -7 a^{-3} z^{-1} +z^5 a^{-1} +6 z^3 a^{-1} +10 z a^{-1} +4 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} -3 z^2 a^{-10} + a^{-10} +z^5 a^{-9} -2 z^3 a^{-9} +z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +z^7 a^{-7} -4 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +z^6 a^{-6} -2 z^4 a^{-6} +3 z^2 a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +7 z a^{-5} -3 a^{-5} z^{-1} +z^8 a^{-4} -9 z^6 a^{-4} +26 z^4 a^{-4} -26 z^2 a^{-4} +7 a^{-4} +z^9 a^{-3} -10 z^7 a^{-3} +34 z^5 a^{-3} -49 z^3 a^{-3} +30 z a^{-3} -7 a^{-3} z^{-1} +z^8 a^{-2} -9 z^6 a^{-2} +25 z^4 a^{-2} -25 z^2 a^{-2} +7 a^{-2} +z^9 a^{-1} -9 z^7 a^{-1} +28 z^5 a^{-1} -37 z^3 a^{-1} +21 z a^{-1} -4 a^{-1} z^{-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16            0
14         21 -1
12        1   1
10       22   0
8     111    -1
6     12     1
4   121      0
2    2       2
0  1         1
-21           1
-41           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $i=6$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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