# L11a391

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a391 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^5+t(2) t(3)^5-t(3)^5-3 t(1) t(3)^4+2 t(1) t(2) t(3)^4-3 t(2) t(3)^4+2 t(3)^4+3 t(1) t(3)^3-2 t(1) t(2) t(3)^3+3 t(2) t(3)^3-2 t(3)^3-3 t(1) t(3)^2+2 t(1) t(2) t(3)^2-3 t(2) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(1) t(2) t(3)+3 t(2) t(3)-2 t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}}$ (db) Jones polynomial $- q^{-14} +2 q^{-13} -6 q^{-12} +9 q^{-11} -12 q^{-10} +15 q^{-9} -14 q^{-8} +14 q^{-7} -8 q^{-6} +7 q^{-5} -3 q^{-4} + q^{-3}$ (db) Signature -6 (db) HOMFLY-PT polynomial $-2 a^{14} z^{-2} -a^{14}+4 a^{12} z^2+7 a^{12} z^{-2} +12 a^{12}-6 a^{10} z^4-23 a^{10} z^2-8 a^{10} z^{-2} -24 a^{10}+3 a^8 z^6+14 a^8 z^4+21 a^8 z^2+3 a^8 z^{-2} +13 a^8+a^6 z^6+3 a^6 z^4+a^6 z^2$ (db) Kauffman polynomial $a^{17} z^5-3 a^{17} z^3+3 a^{17} z-a^{17} z^{-1} +2 a^{16} z^6-3 a^{16} z^4+a^{16}+3 a^{15} z^7-3 a^{15} z^5-2 a^{15} z^3+3 a^{15} z-a^{15} z^{-1} +3 a^{14} z^8-2 a^{14} z^6+2 a^{14} z^4-9 a^{14} z^2-2 a^{14} z^{-2} +7 a^{14}+2 a^{13} z^9+3 a^{13} z^7-13 a^{13} z^5+22 a^{13} z^3-21 a^{13} z+7 a^{13} z^{-1} +a^{12} z^{10}+5 a^{12} z^8-18 a^{12} z^6+33 a^{12} z^4-35 a^{12} z^2-7 a^{12} z^{-2} +22 a^{12}+6 a^{11} z^9-11 a^{11} z^7-8 a^{11} z^5+46 a^{11} z^3-45 a^{11} z+15 a^{11} z^{-1} +a^{10} z^{10}+8 a^{10} z^8-38 a^{10} z^6+62 a^{10} z^4-51 a^{10} z^2-8 a^{10} z^{-2} +28 a^{10}+4 a^9 z^9-8 a^9 z^7-7 a^9 z^5+27 a^9 z^3-24 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-23 a^8 z^6+31 a^8 z^4-24 a^8 z^2-3 a^8 z^{-2} +13 a^8+3 a^7 z^7-8 a^7 z^5+2 a^7 z^3+a^6 z^6-3 a^6 z^4+a^6 z^2$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-5           11
-7          31-2
-9         4  4
-11        43  -1
-13       104   6
-15      77    0
-17     87     1
-19    47      3
-21   58       -3
-23  14        3
-25 15         -4
-27 1          1
-291           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-7$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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