L11n126

L11n125

L11n127

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Link Presentations

[edit Notes on L11n126's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X11,16,12,17 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -11, 9, -10, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{t(1) t(2)^4-t(2)^4-4 t(1) t(2)^3+3 t(2)^3-2 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2+3 t(1)^2 t(2)-4 t(1) t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+5 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+9 q^{3/2}-7 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{2}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-3} -3 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} +2 a z-4 z a^{-1} +4 z a^{-3} -2 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (db)
Kauffman polynomial	$-z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -6 z^8 a^{-4} -3 z^8 a^{-6} -3 z^7 a^{-1} -5 z^7 a^{-3} -5 z^7 a^{-5} -3 z^7 a^{-7} +6 z^6 a^{-2} +15 z^6 a^{-4} +7 z^6 a^{-6} -z^6 a^{-8} -z^6+7 z^5 a^{-1} +20 z^5 a^{-3} +23 z^5 a^{-5} +10 z^5 a^{-7} -8 z^4 a^{-2} -10 z^4 a^{-4} -z^4 a^{-6} +3 z^4 a^{-8} -2 z^4-3 a z^3-15 z^3 a^{-1} -26 z^3 a^{-3} -23 z^3 a^{-5} -9 z^3 a^{-7} +3 z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -2 z^2 a^{-8} +2 z^2+4 a z+11 z a^{-1} +13 z a^{-3} +9 z a^{-5} +3 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               3 1   -2 10             5 2     3 8           4 3       -1 6         6 5         1 4       4 5           1 2     3 5             -2 0   2 5               3 -2   2                 -2 -4 2                   2

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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