# L11a492

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a492 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)^2}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $q^6-4 q^5+9 q^4-15 q^3+20 q^2-22 q+25-19 q^{-1} +15 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -a^2-3 a^{-2} +z^8+5 z^6+10 z^4+9 z^2-2 z^{-2} +3$ (db) Kauffman polynomial $2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +11 a^2 z^8+16 z^8 a^{-2} +7 z^8 a^{-4} +20 z^8+8 a^3 z^7-8 a z^7-24 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-23 a^2 z^6-55 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -66 z^6+a^5 z^5-12 a^3 z^5-5 a z^5-17 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+24 a^2 z^4+60 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +78 z^4-a^5 z^3+4 a^3 z^3+13 a z^3+21 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -13 a^2 z^2-30 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -36 z^2-a^3 z-5 a z-9 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+6 a^{-2} +2 a^{-4} +7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         61 5
7        93  -6
5       116   5
3      119    -2
1     1411     3
-1    915      6
-3   610       -4
-5  39        6
-7 16         -5
-9 3          3
-111           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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