L10n111

L10n110

L10n112

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Link Presentations

[edit Notes on L10n111's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,17,14,20 X19,11,20,16 X7,19,8,18 X15,8,16,9 X9,14,10,15 X17,5,18,10 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {10, -2, -3, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(3) t(4)-1) (t(3) t(1)+t(4) t(1)-t(1)-t(2) t(3)-t(2) t(4)+t(2) t(3) t(4))}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z a^5+2 a^5 z^{-1} +a^5 z^{-3} -2 z^3 a^3-8 z a^3-7 a^3 z^{-1} -3 a^3 z^{-3} +z^5 a+6 z^3 a+10 z a+8 a z^{-1} +3 a z^{-3} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3} -z a^{-3}$ (db)
Kauffman polynomial	$a^5 z^7-6 a^5 z^5+11 a^5 z^3-a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+3 a^4 z^4+8 a^4 z^2+3 a^4 z^{-2} -10 a^4+3 a^3 z^7-20 a^3 z^5+z^5 a^{-3} +39 a^3 z^3-4 z^3 a^{-3} -3 a^3 z^{-3} -31 a^3 z+2 z a^{-3} +12 a^3 z^{-1} +a^2 z^8-4 a^2 z^6+z^6 a^{-2} -6 a^2 z^4-4 z^4 a^{-2} +26 a^2 z^2+2 z^2 a^{-2} +6 a^2 z^{-2} -19 a^2+3 a z^7+z^7 a^{-1} -21 a z^5-6 z^5 a^{-1} +43 a z^3+11 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -35 a z-12 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +2 z^6-13 z^4+20 z^2+3 z^{-2} -10$ (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 3 4 χ 8                     1 1 6                   1 1 0 4               1 1     0 2             1 1 1     1 0           1 4 1       2 -2         2 1 3         4 -4       1 5 1           3 -6     1 1 3             3 -8   1 2                 1 -10                       0 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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