# L11n49

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n49 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^4-v^3+v^2-v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{7}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{2}{q^{19/2}}-\frac{3}{q^{17/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 (-z)-2 a^9 z^{-1} -a^7 z^5-2 a^7 z^3+3 a^7 z+4 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^5-4 a^3 z^3-4 a^3 z-a^3 z^{-1}$ (db) Kauffman polynomial $-3 z^2 a^{12}+2 a^{12}-z^5 a^{11}-z^3 a^{11}-z a^{11}-2 z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}+a^{10}-2 z^7 a^9+z^5 a^9+4 z^3 a^9-4 z a^9+2 a^9 z^{-1} -2 z^8 a^8+4 z^6 a^8-5 z^4 a^8+12 z^2 a^8-6 a^8-z^9 a^7+6 z^5 a^7-z^3 a^7-5 z a^7+4 a^7 z^{-1} -4 z^8 a^6+15 z^6 a^6-17 z^4 a^6+12 z^2 a^6-5 a^6-z^9 a^5+z^7 a^5+9 z^5 a^5-14 z^3 a^5+3 z a^5+a^5 z^{-1} -2 z^8 a^4+9 z^6 a^4-10 z^4 a^4+a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-a^3 z^{-1}$ (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       31 2
-6      33  0
-8     41   3
-10    23    1
-12   44     0
-14  12      1
-16 24       -2
-18 1        1
-202         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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