# L11a494

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a494 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) (w-1)^3 \left(w^2+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $q^3-4 q^2+9 q-12+19 q^{-1} -20 q^{-2} +21 q^{-3} -17 q^{-4} +13 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^6 z^4-3 a^6 z^2-a^6 z^{-2} -3 a^6+2 a^4 z^6+8 a^4 z^4+12 a^4 z^2+4 a^4 z^{-2} +10 a^4-a^2 z^8-5 a^2 z^6-10 a^2 z^4-12 a^2 z^2-5 a^2 z^{-2} -11 a^2+z^6+3 z^4+3 z^2+2 z^{-2} +4$ (db) Kauffman polynomial $a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-4 a^8 z^4+a^8 z^2+6 a^7 z^7-10 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} +7 a^6 z^8-10 a^6 z^6+8 a^6 z^4-9 a^6 z^2-a^6 z^{-2} +5 a^6+5 a^5 z^9+a^5 z^7-20 a^5 z^5+32 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +2 a^4 z^{10}+11 a^4 z^8-38 a^4 z^6+52 a^4 z^4-38 a^4 z^2-4 a^4 z^{-2} +16 a^4+12 a^3 z^9-25 a^3 z^7+8 a^3 z^5+25 a^3 z^3-26 a^3 z+9 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8-50 a^2 z^6+z^6 a^{-2} +65 a^2 z^4-2 z^4 a^{-2} -40 a^2 z^2-5 a^2 z^{-2} +17 a^2+7 a z^9-16 a z^7+4 z^7 a^{-1} +8 a z^5-9 z^5 a^{-1} +5 a z^3+2 z^3 a^{-1} -10 a z+5 a z^{-1} +8 z^8-24 z^6+23 z^4-12 z^2-2 z^{-2} +7$ (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 \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         61 5
1        63  -3
-1       136   7
-3      1110    -1
-5     109     1
-7    711      4
-9   610       -4
-11  27        5
-13 16         -5
-15 2          2
-171           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{13}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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