# L10a173

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### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2) t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-2 t(2) t(4) t(3)^2+2 t(4) t(3)^2-t(3)^2-t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+3 t(1) t(4) t(3)-4 t(1) t(2) t(4) t(3)+3 t(2) t(4) t(3)-4 t(4) t(3)+t(3)+t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+t(1)-2 t(1) t(4)+2 t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $\frac{15}{q^{9/2}}-\frac{17}{q^{7/2}}+\frac{11}{q^{5/2}}-\frac{9}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{16}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z^{-3} +a^9 z+a^9 z^{-1} -3 a^7 z^3-3 a^7 z^{-3} -5 a^7 z-6 a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+3 a^5 z^{-3} +9 a^5 z+9 a^5 z^{-1} +a^3 z^5-a^3 z^3-a^3 z^{-3} -5 a^3 z-4 a^3 z^{-1} -a z^3$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-12 a^9 z^5+14 a^9 z^3-a^9 z^{-3} -12 a^9 z+5 a^9 z^{-1} +5 a^8 z^8-a^8 z^6-13 a^8 z^4+17 a^8 z^2+3 a^8 z^{-2} -10 a^8+2 a^7 z^9+12 a^7 z^7-36 a^7 z^5+42 a^7 z^3-3 a^7 z^{-3} -29 a^7 z+12 a^7 z^{-1} +12 a^6 z^8-17 a^6 z^6-4 a^6 z^4+23 a^6 z^2+6 a^6 z^{-2} -19 a^6+2 a^5 z^9+14 a^5 z^7-40 a^5 z^5+39 a^5 z^3-3 a^5 z^{-3} -23 a^5 z+12 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+7 a^4 z^2+3 a^4 z^{-2} -10 a^4+8 a^3 z^7-16 a^3 z^5+12 a^3 z^3-a^3 z^{-3} -7 a^3 z+5 a^3 z^{-1} +4 a^2 z^6-5 a^2 z^4+a z^5-a z^3$ (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         3 -3
-2        61 5
-4       64  -2
-6      115   6
-8     810    2
-10    87     1
-12   510      5
-14  36       -3
-16  5        5
-1813         -2
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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