# L11a506

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a506 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(3)-1) \left(t(2)^2 t(1)^2+t(2)^2 t(3)^2 t(1)^2-2 t(2) t(3)^2 t(1)^2+t(3)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(3) t(1)^2+2 t(2) t(3) t(1)^2-t(3) t(1)^2-2 t(2)^2 t(1)-t(2)^2 t(3)^2 t(1)+3 t(2) t(3)^2 t(1)-2 t(3)^2 t(1)+3 t(2) t(1)+2 t(2)^2 t(3) t(1)-3 t(2) t(3) t(1)+2 t(3) t(1)-t(1)+t(2)^2-t(2) t(3)^2+t(3)^2-2 t(2)-t(2)^2 t(3)+2 t(2) t(3)-t(3)+1\right)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^7-4 q^6+9 q^5-16 q^4+22 q^3-24 q^2+26 q-21+17 q^{-1} -10 q^{-2} +5 q^{-3} - q^{-4}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-10 z^4 a^{-2} -a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+ a^{-2} +2 z^6+6 z^4+4 z^2-2 z^{-2} -2$ (db) Kauffman polynomial $z^4 a^{-8} +4 z^5 a^{-7} -z^3 a^{-7} +9 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +15 z^7 a^{-5} -20 z^5 a^{-5} +9 z^3 a^{-5} +18 z^8 a^{-4} -34 z^6 a^{-4} +24 z^4 a^{-4} -8 z^2 a^{-4} +13 z^9 a^{-3} +a^3 z^7-17 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +a^3 z^3+10 z^3 a^{-3} +4 z^{10} a^{-2} +5 a^2 z^8+19 z^8 a^{-2} -15 a^2 z^6-77 z^6 a^{-2} +13 a^2 z^4+71 z^4 a^{-2} -2 a^2 z^2-21 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +8 a z^9+21 z^9 a^{-1} -23 a z^7-56 z^7 a^{-1} +16 a z^5+34 z^5 a^{-1} -a z^3-2 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^{10}+6 z^8-49 z^6+53 z^4-14 z^2+2 z^{-2} -3$ (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 \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        103  -7
7       126   6
5      1311    -2
3     1311     2
1    1015      5
-1   711       -4
-3  411        7
-5 16         -5
-7 4          4
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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