# L11n48

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n48 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 \left(-z^3\right)-3 a^9 z-2 a^9 z^{-1} +a^7 z^5+5 a^7 z^3+8 a^7 z+4 a^7 z^{-1} -a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-3 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $a^{12} z^6-5 a^{12} z^4+6 a^{12} z^2-2 a^{12}+a^{11} z^7-4 a^{11} z^5+2 a^{11} z^3+a^{11} z+a^{10} z^8-4 a^{10} z^6+2 a^{10} z^4+2 a^{10} z^2-a^{10}+a^9 z^9-6 a^9 z^7+13 a^9 z^5-15 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +2 a^8 z^8-12 a^8 z^6+24 a^8 z^4-20 a^8 z^2+6 a^8+a^7 z^9-7 a^7 z^7+18 a^7 z^5-22 a^7 z^3+15 a^7 z-4 a^7 z^{-1} +a^6 z^8-7 a^6 z^6+17 a^6 z^4-15 a^6 z^2+5 a^6+a^5 z^5-4 a^5 z^3+5 a^5 z-a^5 z^{-1} +a^4 z^2-a^4+a^3 z^3-3 a^3 z+a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         11
-4        121
-6       1 12
-8      22  0
-10     21   1
-12    131   1
-14   22     0
-16   11     0
-18 12       -1
-20          0
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$

### Computer Talk

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