# L11n385

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n385 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(v-1) (w-1) \left(u w^4-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $q^{-1} - q^{-2} +3 q^{-3} - q^{-4} +3 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10}$ (db) Signature -6 (db) HOMFLY-PT polynomial $-z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+z^6 a^8+6 z^4 a^8+11 z^2 a^8+4 a^8 z^{-2} +10 a^8-z^8 a^6-7 z^6 a^6-17 z^4 a^6-21 z^2 a^6-5 a^6 z^{-2} -16 a^6+z^6 a^4+6 z^4 a^4+11 z^2 a^4+2 a^4 z^{-2} +8 a^4$ (db) Kauffman polynomial $a^{13} z+a^{12} z^2+a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^4-7 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+2 a^9 z^7-11 a^9 z^5+19 a^9 z^3-17 a^9 z+5 a^9 z^{-1} +3 a^8 z^8-19 a^8 z^6+39 a^8 z^4-36 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9-3 a^7 z^7-8 a^7 z^5+30 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-26 a^6 z^6+54 a^6 z^4-47 a^6 z^2-5 a^6 z^{-2} +22 a^6+a^5 z^9-5 a^5 z^7+3 a^5 z^5+12 a^5 z^3-15 a^5 z+5 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+17 a^4 z^4-19 a^4 z^2-2 a^4 z^{-2} +10 a^4$ (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 \
-7-6-5-4-3-2-1012χ
-1         11
-3          0
-5       31 2
-7     112  2
-9     31   2
-11   221    1
-13  143     0
-15 111      1
-17 22       0
-1911        0
-211         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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