L10n59

L10n58

L10n60

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Link Presentations

[edit Notes on L10n59's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, 7, -8, -10, 2, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) (t(2)-t(1)) (t(1) t(2)-1)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a z^5-z^5 a^{-1} -a^3 z^3+6 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -3 a^3 z+11 a z-11 z a^{-1} +3 z a^{-3} -2 a^3 z^{-1} +7 a z^{-1} -7 a^{-1} z^{-1} +2 a^{-3} z^{-1}$ (db)
Kauffman polynomial	$-a^3 z^7-a z^7-z^7 a^{-1} -z^7 a^{-3} -a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} -z^6 a^{-4} -2 z^6+5 a^3 z^5+7 a z^5+7 z^5 a^{-1} +5 z^5 a^{-3} +5 a^4 z^4+12 a^2 z^4+12 z^4 a^{-2} +5 z^4 a^{-4} +14 z^4-6 a^3 z^3-14 a z^3-14 z^3 a^{-1} -6 z^3 a^{-3} -6 a^4 z^2-18 a^2 z^2-18 z^2 a^{-2} -6 z^2 a^{-4} -24 z^2+4 a^3 z+14 a z+14 z a^{-1} +4 z a^{-3} +2 a^4+8 a^2+8 a^{-2} +2 a^{-4} +13-2 a^3 z^{-1} -7 a z^{-1} -7 a^{-1} z^{-1} -2 a^{-3} z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                       0 6                 1 1   0 4             1 1       0 2           1   1       2 0         1 4 1         2 -2       1   1           2 -4       1 1             0 -6   1 1                 0 -8                       0 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

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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