L11n101

L11n100

L11n102

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Link Presentations

[edit Notes on L11n101's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

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

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

Integral Khovanov Homology (db, data source)

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