L10n73

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 Planar diagram presentation X6172 X12,4,13,3 X7,17,8,16 X20,9,11,10 X18,12,19,11 X15,9,16,8 X10,19,5,20 X17,14,18,15 X2536 X4,14,1,13 Gauss code {1, -9, 2, -10}, {9, -1, -3, 6, 4, -7}, {5, -2, 10, 8, -6, 3, -8, -5, 7, -4}
A Braid Representative

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v+w-1) (v w-v-w)}{\sqrt{u} v w}$ (db) Jones polynomial $-q^5+3 q^4-4 q^3- q^{-3} +6 q^2+4 q^{-2} -6 q-4 q^{-1} +7$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $z^5 a^{-5} -2 z^3 a^{-5} +3 z^6 a^{-4} -8 z^4 a^{-4} +4 z^2 a^{-4} +3 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+3 z^3 a^{-3} +z^8 a^{-2} +2 z^6 a^{-2} +4 a^2 z^4-8 z^4 a^{-2} -3 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +a z^7+4 z^7 a^{-1} -8 z^5 a^{-1} +a z^3+5 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8-z^6+4 z^4-2 z^2+2 z^{-2} -3$ (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 \
-3-2-1012345χ
11        1-1
9       2 2
7      21 -1
5     42  2
3    33   0
1   43    1
-1  25     3
-3 22      0
-5 3       3
-71        -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

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