# L11a445

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### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-5 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^8+4 q^7-9 q^6+14 q^5-17 q^4+21 q^3+ q^{-3} -19 q^2-3 q^{-2} +17 q+7 q^{-1} -11$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} -z^2 a^{-6} - a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-3 a^{-2} -2 z^4-3 z^2+ z^{-2}$ (db) Kauffman polynomial $z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -5 z^4 a^{-8} +z^2 a^{-8} +8 z^7 a^{-7} -13 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +9 z^8 a^{-6} -15 z^6 a^{-6} +11 z^4 a^{-6} -7 z^2 a^{-6} +3 a^{-6} +5 z^9 a^{-5} +3 z^7 a^{-5} -20 z^5 a^{-5} +20 z^3 a^{-5} -7 z a^{-5} +z^{10} a^{-4} +16 z^8 a^{-4} -42 z^6 a^{-4} +47 z^4 a^{-4} -31 z^2 a^{-4} - a^{-4} z^{-2} +11 a^{-4} +8 z^9 a^{-3} -6 z^7 a^{-3} -15 z^5 a^{-3} +24 z^3 a^{-3} -12 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +11 z^8 a^{-2} +a^2 z^6-31 z^6 a^{-2} -3 a^2 z^4+35 z^4 a^{-2} +3 a^2 z^2-26 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+11 a^{-2} +3 z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -8 a z^5-17 z^5 a^{-1} +6 a z^3+17 z^3 a^{-1} -a z-8 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-7 z^6+z^4- z^{-2} +3$ (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
17           1-1
15          3 3
13         61 -5
11        83  5
9       107   -3
7      117    4
5     810     2
3    911      -2
1   511       6
-1  26        -4
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{9}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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