L10n50

L10n49

L10n51

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Link Presentations

[edit Notes on L10n50's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X13,20,14,7 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X19,14,20,15 X2,18,3,17
Gauss code	{1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, -4, 9, 7, -8, 10, -2, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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