# L11a455

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a455 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3 w-u v^3+2 u v^2 w^2-4 u v^2 w+2 u v^2+u v w^3-4 u v w^2+3 u v w-u w^3+2 u w^2-2 v^3 w+v^3-3 v^2 w^2+4 v^2 w-v^2-2 v w^3+4 v w^2-2 v w+w^3-w^2}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^7+3 q^6-5 q^5+10 q^4+ q^{-4} -11 q^3-2 q^{-3} +13 q^2+5 q^{-2} -13 q-8 q^{-1} +12$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4-2 z^2 a^2+z^4-2 z^2+ z^{-2} +2 z^4 a^{-2} -2 a^{-2} z^{-2} -3 a^{-2} +z^4 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^2 a^{-6}$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +5 z^8+4 a z^7-14 z^7 a^{-3} -9 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-30 z^6 a^{-2} -29 z^6 a^{-4} -13 z^6 a^{-6} -11 z^6+2 a^3 z^5-4 a z^5-16 z^5 a^{-1} -2 z^5 a^{-3} +4 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-2 a^2 z^4+39 z^4 a^{-2} +39 z^4 a^{-4} +18 z^4 a^{-6} +15 z^4-2 a^3 z^3+2 a z^3+23 z^3 a^{-1} +17 z^3 a^{-3} +2 z^3 a^{-5} +4 z^3 a^{-7} -2 a^4 z^2-29 z^2 a^{-2} -26 z^2 a^{-4} -9 z^2 a^{-6} -10 z^2-11 z a^{-1} -11 z a^{-3} +a^4+13 a^{-2} +8 a^{-4} +5+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-4-3-2-101234567χ
15           1-1
13          2 2
11         31 -2
9        72  5
7       65   -1
5      75    2
3     66     0
1    67      -1
-1   37       4
-3  25        -3
-5  3         3
-712          -1
-91           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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