# L11a46

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a46 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{9/2}-\frac{10}{q^{9/2}}-5 q^{7/2}+\frac{16}{q^{7/2}}+10 q^{5/2}-\frac{22}{q^{5/2}}-17 q^{3/2}+\frac{26}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+22 \sqrt{q}-\frac{26}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z+a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -4 a^3 z-2 a^3 z^{-1} - a^{-3} z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-3 z^3 a^{-1} +3 a z+a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $a^7 z^5-a^7 z^3+4 a^6 z^6-4 a^6 z^4+a^6 z^2+9 a^5 z^7-14 a^5 z^5+11 a^5 z^3-5 a^5 z+a^5 z^{-1} +11 a^4 z^8-15 a^4 z^6+z^6 a^{-4} +9 a^4 z^4-z^4 a^{-4} -3 a^4 z^2+a^4+7 a^3 z^9+7 a^3 z^7+5 z^7 a^{-3} -35 a^3 z^5-10 z^5 a^{-3} +35 a^3 z^3+5 z^3 a^{-3} -15 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^{10}+22 a^2 z^8+9 z^8 a^{-2} -51 a^2 z^6-19 z^6 a^{-2} +36 a^2 z^4+11 z^4 a^{-2} -12 a^2 z^2-2 z^2 a^{-2} +3 a^2+ a^{-2} +14 a z^9+7 z^9 a^{-1} -9 a z^7-2 z^7 a^{-1} -32 a z^5-22 z^5 a^{-1} +36 a z^3+18 z^3 a^{-1} -12 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +2 z^{10}+20 z^8-52 z^6+35 z^4-10 z^2+2$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          4 4
6         61 -5
4        114  7
2       116   -5
0      1511    4
-2     1313     0
-4    913      -4
-6   713       6
-8  39        -6
-10 17         6
-12 3          -3
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{15}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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