# L11a361

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a361 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^3-u^4 v^2+u^3 v^4-4 u^3 v^3+4 u^3 v^2-2 u^3 v-u^2 v^4+4 u^2 v^3-7 u^2 v^2+4 u^2 v-u^2-2 u v^3+4 u v^2-4 u v+u-v^2+v}{u^2 v^2}$ (db) Jones polynomial $-11 q^{9/2}+8 q^{7/2}-5 q^{5/2}+2 q^{3/2}+q^{23/2}-3 q^{21/2}+6 q^{19/2}-10 q^{17/2}+12 q^{15/2}-14 q^{13/2}+13 q^{11/2}-\sqrt{q}$ (db) Signature 5 (db) HOMFLY-PT polynomial $-z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -4 z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +4 z a^{-3} -3 z a^{-7} +2 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-6} -4 z^8 a^{-8} -5 z^8 a^{-10} -z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} -6 z^7 a^{-11} +8 z^6 a^{-4} +8 z^6 a^{-6} +9 z^6 a^{-8} +4 z^6 a^{-10} -5 z^6 a^{-12} +5 z^5 a^{-3} -9 z^5 a^{-7} +6 z^5 a^{-9} +7 z^5 a^{-11} -3 z^5 a^{-13} -9 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +2 z^4 a^{-10} +5 z^4 a^{-12} -z^4 a^{-14} -8 z^3 a^{-3} -3 z^3 a^{-5} +9 z^3 a^{-7} -3 z^3 a^{-9} -4 z^3 a^{-11} +3 z^3 a^{-13} +2 z^2 a^{-4} -2 z^2 a^{-8} -3 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} +4 z a^{-3} -2 z a^{-5} -5 z a^{-7} +2 z a^{-9} -z a^{-13} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-1}$ (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
24           1-1
22          2 2
20         41 -3
18        62  4
16       75   -2
14      75    2
12     67     1
10    57      -2
8   36       3
6  25        -3
4 14         3
2 1          -1
01           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=4$ $i=6$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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