# L11a512

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a512 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^2 t(3)^3-t(2)^2 t(3)^3+t(1)^2 t(2) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-4 t(1) t(2)^2 t(3)^2+2 t(2)^2 t(3)^2-2 t(1) t(3)^2-3 t(1)^2 t(2) t(3)^2+7 t(1) t(2) t(3)^2-3 t(2) t(3)^2+t(3)^2-2 t(1)^2 t(3)-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+4 t(1) t(3)+3 t(1)^2 t(2) t(3)-7 t(1) t(2) t(3)+3 t(2) t(3)-t(3)+t(1)^2-t(1)-t(1)^2 t(2)+2 t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $- q^{-11} +4 q^{-10} -8 q^{-9} +13 q^{-8} -17 q^{-7} +20 q^{-6} -18 q^{-5} +17 q^{-4} -11 q^{-3} +7 q^{-2} -3 q^{-1} +1$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{10} \left(-z^2\right)-a^{10}+3 a^8 z^4+7 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^6-7 a^6 z^4-10 a^6 z^2-2 a^6 z^{-2} -8 a^6-a^4 z^6-a^4 z^4+4 a^4 z^2+a^4 z^{-2} +5 a^4+a^2 z^4+2 a^2 z^2$ (db) Kauffman polynomial $a^{13} z^5-a^{13} z^3+4 a^{12} z^6-6 a^{12} z^4+2 a^{12} z^2+7 a^{11} z^7-11 a^{11} z^5+5 a^{11} z^3-a^{11} z+7 a^{10} z^8-7 a^{10} z^6-a^{10} z^4+a^{10}+4 a^9 z^9+6 a^9 z^7-23 a^9 z^5+18 a^9 z^3-3 a^9 z+a^8 z^{10}+14 a^8 z^8-36 a^8 z^6+39 a^8 z^4-24 a^8 z^2-a^8 z^{-2} +8 a^8+8 a^7 z^9-9 a^7 z^7-6 a^7 z^5+14 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+12 a^6 z^8-40 a^6 z^6+53 a^6 z^4-37 a^6 z^2-2 a^6 z^{-2} +12 a^6+4 a^5 z^9-5 a^5 z^7-3 a^5 z^5+7 a^5 z^3-6 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-14 a^4 z^6+16 a^4 z^4-13 a^4 z^2-a^4 z^{-2} +6 a^4+3 a^3 z^7-8 a^3 z^5+5 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^2 z^2$ (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 \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         51 4
-5        73  -4
-7       104   6
-9      98    -1
-11     119     2
-13    710      3
-15   610       -4
-17  38        5
-19 15         -4
-21 3          3
-231           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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