# L11a389

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a389 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+6 t(1) t(3)^2-5 t(1) t(2) t(3)^2+8 t(2) t(3)^2-5 t(3)^2-8 t(1) t(3)+5 t(1) t(2) t(3)-6 t(2) t(3)+5 t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^5+4 q^4-8 q^3+14 q^2-17 q+19-18 q^{-1} +16 q^{-2} -10 q^{-3} +6 q^{-4} -2 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+3 z^4 a^2+6 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^6-2 z^4-5 z^2-3 z^{-2} -6+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^2 a^{-4}$ (db) Kauffman polynomial $a^2 z^{10}+z^{10}+2 a^3 z^9+6 a z^9+4 z^9 a^{-1} +2 a^4 z^8+5 a^2 z^8+7 z^8 a^{-2} +10 z^8+2 a^5 z^7+3 a^3 z^7-a z^7+5 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6+2 a^4 z^6-a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -11 z^6-5 a^5 z^5-12 a^3 z^5-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-18 a^4 z^4-26 a^2 z^4-7 z^4 a^{-2} -6 z^4 a^{-4} -13 z^4+3 a^5 z^3+8 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+25 a^4 z^2+38 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +24 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 z^{-2}$ (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         51 -4
5        93  6
3       85   -3
1      119    2
-1     1011     1
-3    68      -2
-5   410       6
-7  26        -4
-9 15         4
-11 1          -1
-131           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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