L10n109

L10n108

L10n110

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Link Presentations

[edit Notes on L10n109's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,16,3,15 X16,7,17,8 X19,11,20,14 X13,15,14,20 X9,18,10,19 X11,10,12,5 X17,1,18,4
Gauss code	{1, -4, -3, 10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {4, -5, -10, 8, -6, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-t(2) t(3)^2+t(2) t(4) t(3)^2-t(4) t(3)^2+t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(3)-t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+2 t(4) t(3)-t(3)-t(1) t(4)^2+t(1) t(4)-t(1) t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db)
Jones polynomial	$-2 q^{3/2}+2 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z^3+a^5 z^{-3} +2 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z^{-3} -8 a^3 z-7 a^3 z^{-1} +3 a z^3+3 a z^{-3} - a^{-1} z^{-3} +8 a z+8 a z^{-1} -2 z a^{-1} -3 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-5 a^6 z^4+a^6 z^2+3 a^5 z^7-11 a^5 z^5+15 a^5 z^3-a^5 z^{-3} -12 a^5 z+5 a^5 z^{-1} +a^4 z^8+a^4 z^6-11 a^4 z^4+17 a^4 z^2+3 a^4 z^{-2} -10 a^4+5 a^3 z^7-20 a^3 z^5+36 a^3 z^3-3 a^3 z^{-3} -29 a^3 z+12 a^3 z^{-1} +a^2 z^8-8 a^2 z^4+23 a^2 z^2+6 a^2 z^{-2} -19 a^2+2 a z^7-8 a z^5+21 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -23 a z-7 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +z^6-2 z^4+7 z^2+3 z^{-2} -10$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 4                 2 2 2               1 1 0 0             5 1   4 -2           3 5     2 -4         4 1       3 -6       2 5         3 -8     2 2           0 -10     2             2 -12 1 2               -1 -14 1                 1

Integral Khovanov Homology (db, data source)

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

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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