# L10a166

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### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(4)^2 t(3)^2-t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2+t(1) t(4) t(3)^2-t(1) t(2) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(4)^2 t(3)-t(4)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-3 t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-3 t(2) t(4) t(3)+2 t(4) t(3)-t(3)-t(1)+t(1) t(2)-t(2)+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{2}{q^{23/2}}-\frac{6}{q^{21/2}}+\frac{8}{q^{19/2}}-\frac{12}{q^{17/2}}+\frac{11}{q^{15/2}}-\frac{13}{q^{13/2}}+\frac{8}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{13} z^{-1} +a^{13} z^{-3} -4 z a^{11}-8 a^{11} z^{-1} -3 a^{11} z^{-3} +6 z^3 a^9+17 z a^9+13 a^9 z^{-1} +3 a^9 z^{-3} -3 z^5 a^7-11 z^3 a^7-13 z a^7-6 a^7 z^{-1} -a^7 z^{-3} -z^5 a^5-2 z^3 a^5$ (db) Kauffman polynomial $-z^5 a^{15}+3 z^3 a^{15}-3 z a^{15}+a^{15} z^{-1} -2 z^6 a^{14}+3 z^4 a^{14}-a^{14}-3 z^7 a^{13}+4 z^5 a^{13}-2 z^3 a^{13}+3 z a^{13}-3 a^{13} z^{-1} +a^{13} z^{-3} -2 z^8 a^{12}-4 z^6 a^{12}+15 z^4 a^{12}-16 z^2 a^{12}-3 a^{12} z^{-2} +11 a^{12}-z^9 a^{11}-6 z^7 a^{11}+15 z^5 a^{11}-20 z^3 a^{11}+21 z a^{11}-12 a^{11} z^{-1} +3 a^{11} z^{-3} -6 z^8 a^{10}+5 z^6 a^{10}+14 z^4 a^{10}-33 z^2 a^{10}-6 a^{10} z^{-2} +24 a^{10}-z^9 a^9-9 z^7 a^9+28 z^5 a^9-37 z^3 a^9+28 z a^9-14 a^9 z^{-1} +3 a^9 z^{-3} -4 z^8 a^8+4 z^6 a^8+7 z^4 a^8-17 z^2 a^8-3 a^8 z^{-2} +13 a^8-6 z^7 a^7+17 z^5 a^7-20 z^3 a^7+13 z a^7-6 a^7 z^{-1} +a^7 z^{-3} -3 z^6 a^6+5 z^4 a^6-z^5 a^5+2 z^3 a^5$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-4          11
-6         31-2
-8        4  4
-10       43  -1
-12      94   5
-14     57    2
-16    76     1
-18   48      4
-20  24       -2
-22  4        4
-2412         -1
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{7}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{9}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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