# L11n396

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n396 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $0$ (db) Jones polynomial $q^6-2 q^5+2 q^4-2 q^3+q^2+q+1+3 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^6 a^{-2} +z^6-a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +6 z^4-3 a^2 z^2-11 z^2 a^{-2} +3 z^2 a^{-4} +11 z^2-2 a^2-6 a^{-2} +2 a^{-4} +6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $a z^9+2 z^9 a^{-1} +z^9 a^{-3} +2 a^2 z^8+5 z^8 a^{-2} +2 z^8 a^{-4} +5 z^8+a^3 z^7-3 a z^7-9 z^7 a^{-1} -3 z^7 a^{-3} +2 z^7 a^{-5} -11 a^2 z^6-33 z^6 a^{-2} -10 z^6 a^{-4} +z^6 a^{-6} -33 z^6-5 a^3 z^5-8 a z^5-4 z^5 a^{-1} -10 z^5 a^{-3} -9 z^5 a^{-5} +17 a^2 z^4+59 z^4 a^{-2} +11 z^4 a^{-4} -4 z^4 a^{-6} +61 z^4+6 a^3 z^3+20 a z^3+32 z^3 a^{-1} +26 z^3 a^{-3} +8 z^3 a^{-5} -14 a^2 z^2-40 z^2 a^{-2} -8 z^2 a^{-4} +2 z^2 a^{-6} -44 z^2-2 a^3 z-10 a z-18 z a^{-1} -14 z a^{-3} -4 z a^{-5} +4 a^2+12 a^{-2} +4 a^{-4} +13-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-10123456χ
13           11
11          1 -1
9        121 0
7       121  0
5      232   -1
3     332    2
1    262     2
-1   235      4
-3  12        1
-5 121        0
-7 1          1
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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