# L11a448

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a448 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^9+3 q^8-5 q^7+7 q^6-9 q^5+11 q^4-9 q^3+9 q^2+ q^{-2} -6 q-2 q^{-1} +5$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-6} -4 z^4 a^{-6} -4 z^2 a^{-6} -2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +13 z^4 a^{-4} +14 z^2 a^{-4} + a^{-4} z^{-2} +7 a^{-4} -2 z^6 a^{-2} -10 z^4 a^{-2} -15 z^2 a^{-2} -2 a^{-2} z^{-2} -9 a^{-2} +z^4+4 z^2+ z^{-2} +4$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +z^8 a^{-2} +6 z^8 a^{-4} +6 z^8 a^{-6} +z^8-10 z^7 a^{-1} -25 z^7 a^{-3} -9 z^7 a^{-5} +6 z^7 a^{-7} -25 z^6 a^{-2} -39 z^6 a^{-4} -14 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+14 z^5 a^{-1} +24 z^5 a^{-3} -3 z^5 a^{-5} -8 z^5 a^{-7} +5 z^5 a^{-9} +54 z^4 a^{-2} +56 z^4 a^{-4} +5 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-3 z^3 a^{-1} +4 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -41 z^2 a^{-2} -31 z^2 a^{-4} +2 z^2 a^{-8} -z^2 a^{-10} -13 z^2-5 z a^{-1} -8 z a^{-3} -3 z a^{-5} +z a^{-7} +z a^{-9} +13 a^{-2} +9 a^{-4} - a^{-8} +6+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-4-3-2-101234567χ
19           1-1
17          2 2
15         31 -2
13        42  2
11       64   -2
9      53    2
7     46     2
5    55      0
3   47       3
1  12        -1
-1 14         3
-3 1          -1
-51           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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