# L11a395

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a395 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v w^3-4 u v w^2+4 u v w-2 u v-2 u w^3+5 u w^2-5 u w+2 u-2 v w^3+5 v w^2-5 v w+2 v+2 w^3-4 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $1-4 q^{-1} +8 q^{-2} -11 q^{-3} +15 q^{-4} -15 q^{-5} +17 q^{-6} -12 q^{-7} +9 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}$ (db) Signature -4 (db) HOMFLY-PT polynomial $-a^{12} z^{-2} +4 a^{10} z^{-2} +4 a^{10}-6 z^2 a^8-5 a^8 z^{-2} -11 a^8+4 z^4 a^6+9 z^2 a^6+2 a^6 z^{-2} +7 a^6-z^6 a^4-2 z^4 a^4-z^2 a^4+z^4 a^2+z^2 a^2$ (db) Kauffman polynomial $z^5 a^{13}-3 z^3 a^{13}+3 z a^{13}-a^{13} z^{-1} +2 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+a^{12} z^{-2} -2 a^{12}+2 z^7 a^{11}+2 z^5 a^{11}-12 z^3 a^{11}+13 z a^{11}-5 a^{11} z^{-1} +2 z^8 a^{10}+4 z^6 a^{10}-13 z^4 a^{10}+14 z^2 a^{10}+4 a^{10} z^{-2} -10 a^{10}+2 z^9 a^9+z^7 a^9+3 z^5 a^9-16 z^3 a^9+21 z a^9-9 a^9 z^{-1} +z^{10} a^8+4 z^8 a^8-4 z^6 a^8-8 z^4 a^8+20 z^2 a^8+5 a^8 z^{-2} -14 a^8+6 z^9 a^7-9 z^7 a^7+4 z^5 a^7-6 z^3 a^7+11 z a^7-5 a^7 z^{-1} +z^{10} a^6+8 z^8 a^6-22 z^6 a^6+11 z^4 a^6+6 z^2 a^6+2 a^6 z^{-2} -7 a^6+4 z^9 a^5-4 z^7 a^5-8 z^5 a^5+6 z^3 a^5+6 z^8 a^4-15 z^6 a^4+8 z^4 a^4-2 z^2 a^4+4 z^7 a^3-10 z^5 a^3+5 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^2$ (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          3 -3
-3         51 4
-5        74  -3
-7       84   4
-9      77    0
-11     108     2
-13    510      5
-15   47       -3
-17  15        4
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{10}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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