# L11a476

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a476 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1)^2 (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^5+4 q^4-10 q^3+16 q^2-20 q+25-22 q^{-1} +20 q^{-2} -14 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +11 z^4+3 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +14 z^2+3 a^4-10 a^2-3 a^{-2} +10+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-z^3 a^{-5} -a^5 z+5 a^4 z^8-9 a^4 z^6+4 z^6 a^{-4} +4 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-4 a^3 z^7+9 z^7 a^{-3} -7 a^3 z^5-14 z^5 a^{-3} +9 a^3 z^3+8 z^3 a^{-3} -7 a^3 z-3 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+12 z^8 a^{-2} -38 a^2 z^6-23 z^6 a^{-2} +43 a^2 z^4+21 z^4 a^{-2} -31 a^2 z^2-12 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+5 a^{-2} +13 a z^9+8 z^9 a^{-1} -20 a z^7-4 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +15 a z^3+20 z^3 a^{-1} -13 a z-10 z a^{-1} +2 a z^{-1} +2 z^{10}+19 z^8-55 z^6+61 z^4-38 z^2- z^{-2} +15$ (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          3 3
7         71 -6
5        93  6
3       117   -4
1      149    5
-1     1013     3
-3    1012      -2
-5   612       6
-7  28        -6
-9 16         5
-11 2          -2
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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