# L11n1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n1 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^5-4 t(1) t(2)^4+2 t(1) t(2)^3+2 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +a^9 z^3+5 a^9 z+3 a^9 z^{-1} -2 a^7 z^5-9 a^7 z^3-9 a^7 z-3 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $-z^2 a^{12}+a^{12}-2 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^6 a^{10}+3 z^4 a^{10}-5 z^2 a^{10}+2 a^{10}-2 z^7 a^9+9 z^5 a^9-16 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^8 a^8+8 z^6 a^8-9 z^4 a^8+4 z^2 a^8-z^9 a^7+z^7 a^7+12 z^5 a^7-25 z^3 a^7+17 z a^7-3 a^7 z^{-1} -4 z^8 a^6+19 z^6 a^6-25 z^4 a^6+12 z^2 a^6-2 a^6-z^9 a^5+2 z^7 a^5+8 z^5 a^5-18 z^3 a^5+11 z a^5-2 a^5 z^{-1} -2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+4 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+4 z a^3-a^3 z^{-1}$ (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
0         11
-2        1 -1
-4       21 1
-6     132  0
-8     31   2
-10   133    1
-12  143     0
-14  12      1
-16 23       -1
-18 1        1
-201         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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