# L11a493

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a493 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1) \left(t(3)^2-t(3)+1\right) \left(t(2) t(3)^2-2 t(2) t(3)+2 t(3)-1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}}$ (db) Jones polynomial $q^{10}-4 q^9+10 q^8-16 q^7+20 q^6-23 q^5+24 q^4-18 q^3+15 q^2-8 q+4- q^{-1}$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^8 a^{-4} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +9 z^2 a^{-4} -9 z^2 a^{-6} +2 z^2 a^{-8} + a^{-2} +2 a^{-4} -5 a^{-6} +2 a^{-8} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db) Kauffman polynomial $2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +14 z^9 a^{-5} +9 z^9 a^{-7} +4 z^8 a^{-2} +12 z^8 a^{-4} +24 z^8 a^{-6} +16 z^8 a^{-8} +z^7 a^{-1} -11 z^7 a^{-3} -26 z^7 a^{-5} +2 z^7 a^{-7} +16 z^7 a^{-9} -14 z^6 a^{-2} -56 z^6 a^{-4} -76 z^6 a^{-6} -24 z^6 a^{-8} +10 z^6 a^{-10} -3 z^5 a^{-1} -2 z^5 a^{-3} -11 z^5 a^{-5} -39 z^5 a^{-7} -23 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-2} +66 z^4 a^{-4} +66 z^4 a^{-6} +8 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +14 z^3 a^{-3} +32 z^3 a^{-5} +34 z^3 a^{-7} +13 z^3 a^{-9} -8 z^2 a^{-2} -29 z^2 a^{-4} -25 z^2 a^{-6} +4 z^2 a^{-10} -z a^{-1} -3 z a^{-3} -11 z a^{-5} -13 z a^{-7} -4 z a^{-9} +4 a^{-4} +5 a^{-6} + a^{-8} - a^{-10} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (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 \
-3-2-1012345678χ
21           11
19          3 -3
17         71 6
15        93  -6
13       117   4
11      1411    -3
9     109     1
7    814      6
5   710       -3
3  310        7
1 15         -4
-1 3          3
-31           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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