L10n47

L10n46

L10n48

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Link Presentations

[edit Notes on L10n47's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X14,20,15,19 X17,2,18,3
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 $u$, $v$, $w$, ...)	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 -\frac{-u^2 v^2+2 u^2 v+u v^4-2 u v^3+3 u v^2-2 u v+u+2 v^3-v^2}{u v^2}} (db)
Jones polynomial	$-q^{3/2}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-7za^{3}-3a^{3}z^{-1}+2z^{3}a+3za+az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-3a^{5}z^{7}-4a^{3}z^{7}-az^{7}-2a^{6}z^{6}+2a^{2}z^{6}-a^{7}z^{5}+11a^{5}z^{5}+14a^{3}z^{5}+2az^{5}+5a^{6}z^{4}+6a^{4}z^{4}-2a^{2}z^{4}-3z^{4}+3a^{7}z^{3}-14a^{5}z^{3}-21a^{3}z^{3}-5az^{3}-z^{3}a^{-1}-a^{6}z^{2}-8a^{4}z^{2}-4a^{2}z^{2}+3z^{2}-a^{7}z+9a^{5}z+12a^{3}z+3az+za^{-1}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 4                 1 1 2               2   -2 0             2 1   1 -2           4 3     -1 -4         2 1       1 -6       2 4         2 -8     2 2           0 -10     2             2 -12 1 2               -1 -14 1                 1

Integral Khovanov Homology (db, data source)

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

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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