L11n145

L11n144

L11n146

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Link Presentations

[edit Notes on L11n145's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X15,20,16,21 X13,22,14,7 X21,14,22,15 X4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, -9, 10, -8, -7, 4, -2, 11, 8, -10, 9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, 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 w} , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-4a^{3}z^{5}+az^{5}+2a^{5}z^{3}-4a^{3}z^{3}+2az^{3}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-4z^{4}a^{8}+2z^{2}a^{8}-z^{7}a^{7}-3z^{5}a^{7}+2z^{3}a^{7}-3z^{8}a^{6}+7z^{6}a^{6}-12z^{4}a^{6}+6z^{2}a^{6}-2z^{9}a^{5}+2z^{7}a^{5}-8z^{8}a^{4}+23z^{6}a^{4}-20z^{4}a^{4}+6z^{2}a^{4}-2z^{9}a^{3}-z^{7}a^{3}+15z^{5}a^{3}-10z^{3}a^{3}-za^{3}+a^{3}z^{-1}-5z^{8}a^{2}+15z^{6}a^{2}-10z^{4}a^{2}+2z^{2}a^{2}-a^{2}-4z^{7}a+12z^{5}a-7z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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 3 χ 4                   1 -1 2                 3   3 0               3 1   -2 -2             6 3     3 -4           5 4       -1 -6         6 5         1 -8       4 6           2 -10     3 5             -2 -12   1 4               3 -14   3                 -3 -16 1                   1

Integral Khovanov Homology (db, data source)

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