# L11n391

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n391 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $1-3 q^{-1} +6 q^{-2} -5 q^{-3} +7 q^{-4} -5 q^{-5} +5 q^{-6} -2 q^{-7} + q^{-9} - q^{-10}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{10} z^{-2} -a^{10}+2 a^8 z^2+4 a^8 z^{-2} +6 a^8-4 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-3 a^4 z^4+2 a^4 z^{-2} +4 a^4+a^2 z^4+2 a^2 z^2+a^2$ (db) Kauffman polynomial $a^{11} z^7-6 a^{11} z^5+9 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+12 a^{10} z^4-8 a^{10} z^2-a^{10} z^{-2} +3 a^{10}+a^9 z^7-12 a^9 z^5+29 a^9 z^3-20 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-15 a^8 z^6+33 a^8 z^4-25 a^8 z^2-4 a^8 z^{-2} +13 a^8+a^7 z^9-2 a^7 z^7-11 a^7 z^5+35 a^7 z^3-30 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-17 a^6 z^6+25 a^6 z^4-23 a^6 z^2-5 a^6 z^{-2} +16 a^6+a^5 z^9+a^5 z^7-14 a^5 z^5+19 a^5 z^3-15 a^5 z+5 a^5 z^{-1} +3 a^4 z^8-8 a^4 z^6+a^4 z^4-3 a^4 z^2-2 a^4 z^{-2} +6 a^4+3 a^3 z^7-9 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+3 a^2 z^2-a^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         41 3
-5        34  1
-7      152   2
-9     123    2
-11    165     0
-13   113      3
-15   13       -2
-17 111        1
-19            0
-211           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{6}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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