# L11n361

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n361 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (w-1)^2 \left(v^2+v w+w^2\right)}{\sqrt{u} v w^2}$ (db) Jones polynomial $- q^{-10} +3 q^{-9} -4 q^{-8} +7 q^{-7} -7 q^{-6} +8 q^{-5} -7 q^{-4} +6 q^{-3} -3 q^{-2} +2 q^{-1}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^{10}+3 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^4-5 a^6 z^2-2 a^6 z^{-2} -6 a^6-a^4 z^4+a^4 z^{-2} +a^4+2 a^2 z^2+2 a^2$ (db) Kauffman polynomial $z^7 a^{11}-4 z^5 a^{11}+4 z^3 a^{11}-z a^{11}+3 z^8 a^{10}-14 z^6 a^{10}+19 z^4 a^{10}-8 z^2 a^{10}+2 z^9 a^9-5 z^7 a^9-5 z^5 a^9+12 z^3 a^9-2 z a^9+8 z^8 a^8-35 z^6 a^8+47 z^4 a^8-27 z^2 a^8-a^8 z^{-2} +7 a^8+2 z^9 a^7-2 z^7 a^7-12 z^5 a^7+17 z^3 a^7-9 z a^7+2 a^7 z^{-1} +5 z^8 a^6-19 z^6 a^6+28 z^4 a^6-25 z^2 a^6-2 a^6 z^{-2} +11 a^6+4 z^7 a^5-10 z^5 a^5+10 z^3 a^5-7 z a^5+2 a^5 z^{-1} +2 z^6 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2$ (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 \
-9-8-7-6-5-4-3-2-10χ
-1         22
-3        32-1
-5       3  3
-7      43  -1
-9     43   1
-11    34    1
-13   44     0
-15  25      3
-17 12       -1
-19 2        2
-211         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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