L10n27

L10n26

L10n28

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Link Presentations

[edit Notes on L10n27's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X17,5,18,20 X13,18,14,19 X19,12,20,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, -3, 8, -7, -4, 5, 3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 2                 1 1 0               2   -2 -2             3 1   2 -4           3 3     0 -6         4 2       2 -8       2 3         1 -10     3 4           -1 -12   1 3             2 -14   2               -2 -16 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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