# L11n445

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n445 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(3)-1) (t(4)-1) \left(t(4)^2+t(1) t(4)-2 t(1) t(2) t(4)+t(2) t(4)-2 t(4)+t(1) t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}}$ (db) Jones polynomial $-10 q^{9/2}+9 q^{7/2}-12 q^{5/2}+9 q^{3/2}-\frac{2}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-9 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-7} + a^{-7} z^{-1} -3 z^3 a^{-5} - a^{-5} z^{-3} -6 z a^{-5} -5 a^{-5} z^{-1} +2 z^5 a^{-3} +7 z^3 a^{-3} +3 a^{-3} z^{-3} +12 z a^{-3} +10 a^{-3} z^{-1} -4 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +2 a z-9 z a^{-1} +3 a z^{-1} -9 a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-3} -z^9 a^{-5} -4 z^8 a^{-2} -7 z^8 a^{-4} -3 z^8 a^{-6} -4 z^7 a^{-1} -10 z^7 a^{-3} -9 z^7 a^{-5} -3 z^7 a^{-7} +8 z^6 a^{-2} +13 z^6 a^{-4} +3 z^6 a^{-6} -z^6 a^{-8} -z^6+14 z^5 a^{-1} +43 z^5 a^{-3} +38 z^5 a^{-5} +9 z^5 a^{-7} +9 z^4 a^{-4} +12 z^4 a^{-6} +3 z^4 a^{-8} -3 a z^3-33 z^3 a^{-1} -66 z^3 a^{-3} -45 z^3 a^{-5} -9 z^3 a^{-7} -24 z^2 a^{-2} -30 z^2 a^{-4} -14 z^2 a^{-6} -3 z^2 a^{-8} -5 z^2+7 a z+30 z a^{-1} +48 z a^{-3} +31 z a^{-5} +6 z a^{-7} +21 a^{-2} +18 a^{-4} +6 a^{-6} + a^{-8} +9-5 a z^{-1} -14 a^{-1} z^{-1} -18 a^{-3} z^{-1} -11 a^{-5} z^{-1} -2 a^{-7} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3}$ (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 \
-2-101234567χ
16         1-1
14        2 2
12       41 -3
10      62  4
8     56   1
6    74    3
4  135     3
2  77      0
0 28       6
-2 1        -1
-42         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{7}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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