# L11a381

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a381 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^2 t(1)^4-t(2) t(1)^4-t(2)^4 t(1)^3+5 t(2)^3 t(1)^3-8 t(2)^2 t(1)^3+5 t(2) t(1)^3-t(1)^3+2 t(2)^4 t(1)^2-9 t(2)^3 t(1)^2+15 t(2)^2 t(1)^2-9 t(2) t(1)^2+2 t(1)^2-t(2)^4 t(1)+5 t(2)^3 t(1)-8 t(2)^2 t(1)+5 t(2) t(1)-t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{4}{q^{7/2}}-23 q^{5/2}+\frac{10}{q^{5/2}}+26 q^{3/2}-\frac{17}{q^{3/2}}-q^{13/2}+4 q^{11/2}-27 \sqrt{q}+\frac{22}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3+3 z^3 a^{-3} -z^3 a^{-5} -a^3 z+3 z a^{-1} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +12 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -2 a^4 z^4+12 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} +9 z^9 a^{-3} +4 a^3 z^7-3 z^7 a^{-3} -8 a^3 z^5-15 z^5 a^{-3} +5 a^3 z^3+13 z^3 a^{-3} -a^3 z-z a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +8 a^2 z^8+18 z^8 a^{-2} -17 a^2 z^6-46 z^6 a^{-2} +12 a^2 z^4+33 z^4 a^{-2} -4 a^2 z^2-9 z^2 a^{-2} + a^{-2} +8 a z^9+17 z^9 a^{-1} -11 a z^7-27 z^7 a^{-1} +7 z^5 a^{-1} -2 z^3 a^{-1} +2 a z+5 z a^{-1} - a^{-1} z^{-1} +3 z^{10}+14 z^8-42 z^6+31 z^4-9 z^2$ (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 \
-5-4-3-2-10123456χ
14           11
12          3 -3
10         71 6
8        103  -7
6       137   6
4      1411    -3
2     1312     1
0    1015      5
-2   712       -5
-4  310        7
-6 17         -6
-8 3          3
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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