# L11a205

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a205 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-2 t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-2 t(1)+2}{t(1) t(2)^2}$ (db) Jones polynomial $-\frac{9}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{11}{q^{15/2}}-\frac{12}{q^{13/2}}+\frac{11}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-z^5 a^9-3 z^3 a^9-2 z a^9+z^7 a^7+4 z^5 a^7+4 z^3 a^7-a^7 z^{-1} +z^7 a^5+5 z^5 a^5+9 z^3 a^5+8 z a^5+3 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-7 z a^3-2 a^3 z^{-1}$ (db) Kauffman polynomial $-z^4 a^{14}+z^2 a^{14}-3 z^5 a^{13}+3 z^3 a^{13}-5 z^6 a^{12}+6 z^4 a^{12}-2 z^2 a^{12}-6 z^7 a^{11}+10 z^5 a^{11}-8 z^3 a^{11}+z a^{11}-5 z^8 a^{10}+9 z^6 a^{10}-9 z^4 a^{10}+3 z^2 a^{10}-3 z^9 a^9+5 z^7 a^9-7 z^5 a^9+8 z^3 a^9-3 z a^9-z^{10} a^8-z^8 a^8+3 z^6 a^8+2 z^4 a^8-2 z^2 a^8+a^8-4 z^9 a^7+13 z^7 a^7-18 z^5 a^7+15 z^3 a^7-2 z a^7-a^7 z^{-1} -z^{10} a^6+3 z^8 a^6-8 z^6 a^6+19 z^4 a^6-15 z^2 a^6+3 a^6-z^9 a^5+z^7 a^5+8 z^5 a^5-16 z^3 a^5+11 z a^5-3 a^5 z^{-1} -z^8 a^4+3 z^6 a^4+z^4 a^4-7 z^2 a^4+3 a^4-z^7 a^3+6 z^5 a^3-12 z^3 a^3+9 z a^3-2 a^3 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-1012χ
0           11
-2            0
-4         41 3
-6        31  -2
-8       63   3
-10      64    -2
-12     65     1
-14    56      1
-16   46       -2
-18  25        3
-20 14         -3
-22 2          2
-241           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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