L10n104

L10n103

L10n105

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Link Presentations

[edit Notes on L10n104's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code	{1, 4, -3, -10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {-4, 5, 10, 9, -8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1) t(2)-t(3) t(4)) (t(3) t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -2 a^7 z-6 a^7 z^{-1} +a^5 z^3+3 a^5 z^{-3} +6 a^5 z+9 a^5 z^{-1} -a^3 z^3-a^3 z^{-3} -4 a^3 z-4 a^3 z^{-1}$ (db)
Kauffman polynomial	$a^9 z^7-7 a^9 z^5+15 a^9 z^3-a^9 z^{-3} -12 a^9 z+5 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+3 a^8 z^{-2} -10 a^8+a^7 z^7-8 a^7 z^5+21 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^6-8 a^6 z^4+20 a^6 z^2+6 a^6 z^{-2} -19 a^6-a^5 z^5+7 a^5 z^3-3 a^5 z^{-3} -15 a^5 z+12 a^5 z^{-1} -a^4 z^4+6 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^3-a^3 z^{-3} -4 a^3 z+5 a^3 z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 1 1 -4             1   1 2 -6           1 4     3 -8             3     3 -10       1 3         2 -12         2         2 -14     1             1 -16 1                 1 -18 1                 1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

Computer Talk

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

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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