# L11a360

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a360 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^4-u^4 v^3-u^3 v^4+u^3 v^3-u^3 v^2-u^2 v^3+u^2 v^2-u^2 v-u v^2+u v-u-v+1}{u^2 v^2}$ (db) Jones polynomial $2 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}+q^{25/2}-2 q^{23/2}+2 q^{21/2}-3 q^{19/2}+3 q^{17/2}-3 q^{15/2}+3 q^{13/2}-3 q^{11/2}$ (db) Signature 7 (db) HOMFLY-PT polynomial $-z^9 a^{-7} +z^7 a^{-5} -8 z^7 a^{-7} +z^7 a^{-9} +7 z^5 a^{-5} -22 z^5 a^{-7} +6 z^5 a^{-9} +15 z^3 a^{-5} -25 z^3 a^{-7} +10 z^3 a^{-9} +10 z a^{-5} -11 z a^{-7} +4 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-6} -z^{10} a^{-8} -z^9 a^{-5} -3 z^9 a^{-7} -2 z^9 a^{-9} +7 z^8 a^{-6} +5 z^8 a^{-8} -2 z^8 a^{-10} +8 z^7 a^{-5} +21 z^7 a^{-7} +11 z^7 a^{-9} -2 z^7 a^{-11} -15 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -2 z^6 a^{-12} -22 z^5 a^{-5} -48 z^5 a^{-7} -18 z^5 a^{-9} +6 z^5 a^{-11} -2 z^5 a^{-13} +9 z^4 a^{-6} -3 z^4 a^{-8} -6 z^4 a^{-10} +4 z^4 a^{-12} -2 z^4 a^{-14} +25 z^3 a^{-5} +42 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} +2 z^3 a^{-13} -2 z^3 a^{-15} +2 z^2 a^{-6} +3 z^2 a^{-8} -z^2 a^{-10} +z^2 a^{-14} -z^2 a^{-16} -11 z a^{-5} -13 z a^{-7} -3 z a^{-9} -z a^{-11} +z a^{-13} +z a^{-15} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-1}$ (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 \
-2-10123456789χ
26           1-1
24          1 1
22         11 0
20        21  1
18       22   0
16      11    0
14     22     0
12    11      0
10   12       1
8  11        0
6 12         1
4            0
21           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=6$ $i=8$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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