# L11a449

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a449 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-4 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+5 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-5 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^5+3 q^4-6 q^3+10 q^2-13 q+17-15 q^{-1} +14 q^{-2} -10 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +14 z^4+3 a^4 z^2-14 a^2 z^2-5 z^2 a^{-2} +15 z^2+3 a^4-9 a^2-2 a^{-2} +8+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-2 z^3 a^{-5} -a^5 z+4 a^4 z^8-8 a^4 z^6+3 z^6 a^{-4} +a^4 z^4-6 z^4 a^{-4} -2 a^4 z^2+z^2 a^{-4} -a^4 z^{-2} +3 a^4+3 a^3 z^9-a^3 z^7+5 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +14 a^3 z^3+7 z^3 a^{-3} -8 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -30 a^2 z^6-16 z^6 a^{-2} +39 a^2 z^4+21 z^4 a^{-2} -28 a^2 z^2-13 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2+3 a^{-2} +7 a z^9+4 z^9 a^{-1} -14 a z^7-5 z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +16 a z^3+16 z^3 a^{-1} -12 a z-7 z a^{-1} +2 a z^{-1} +z^{10}+10 z^8-40 z^6+62 z^4-37 z^2- z^{-2} +11$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         41 -3
5        62  4
3       74   -3
1      106    4
-1     810     2
-3    67      -1
-5   48       4
-7  36        -3
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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