# L11a475

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a475 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(2)-1) (t(3)-1)^2 (t(3) t(1)-3 t(1)-3 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^5+4 q^4-9 q^3+15 q^2-19 q+21-20 q^{-1} +18 q^{-2} -11 q^{-3} +7 q^{-4} -2 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -z^2 a^{-4} -4 a^4+3 a^2 z^4+2 z^4 a^{-2} +4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2-z^6-z^4-z^2$ (db) Kauffman polynomial $a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +3 a^4 z^8+11 a^2 z^8+9 z^8 a^{-2} +17 z^8+2 a^5 z^7-3 a z^7+7 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-3 a^4 z^6-27 a^2 z^6-11 z^6 a^{-2} +4 z^6 a^{-4} -38 z^6-4 a^5 z^5-7 a^3 z^5-18 a z^5-28 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-5 a^4 z^4+22 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +31 z^4+4 a^3 z^3+18 a z^3+22 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+9 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +2 z^2 a^{-4} -14 z^2+4 a^5 z+2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} -4 a^6-6 a^4-a^2+ a^{-2} +3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2}$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9          3 3
7         61 -5
5        93  6
3       106   -4
1      119    2
-1     1112     1
-3    79      -2
-5   411       7
-7  37        -4
-9 16         5
-11 1          -1
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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