L11a316

L11a315

L11a317

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Link Presentations

[edit Notes on L11a316's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X6,9,7,10 X16,7,17,8 X8,15,1,16 X20,13,21,14 X18,6,19,5 X4,20,5,19 X22,17,9,18 X14,21,15,22
Gauss code	{1, -2, 3, -9, 8, -4, 5, -6}, {4, -1, 2, -3, 7, -11, 6, -5, 10, -8, 9, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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} , ...)	${\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+5u^{2}v^{3}-5u^{2}v^{2}+3u^{2}v-u^{2}-uv^{5}+3uv^{4}-5uv^{3}+5uv^{2}-4uv+u-v^{4}+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}+{\frac {10}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {6}{q^{19/2}}}-{\frac {10}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {16}{q^{13/2}}}+{\frac {15}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-3a^{9}z^{3}-2a^{9}z-a^{9}z^{-1}+a^{7}z^{7}+4a^{7}z^{5}+6a^{7}z^{3}+6a^{7}z+3a^{7}z^{-1}+a^{5}z^{7}+3a^{5}z^{5}-4a^{5}z-2a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-3a^{13}z^{3}+5a^{12}z^{6}-5a^{12}z^{4}+a^{12}z^{2}+7a^{11}z^{7}-11a^{11}z^{5}+8a^{11}z^{3}-2a^{11}z+8a^{10}z^{8}-18a^{10}z^{6}+18a^{10}z^{4}-6a^{10}z^{2}+a^{10}+6a^{9}z^{9}-12a^{9}z^{7}+6a^{9}z^{5}+a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+2a^{8}z^{10}+6a^{8}z^{8}-33a^{8}z^{6}+39a^{8}z^{4}-18a^{8}z^{2}+3a^{8}+10a^{7}z^{9}-33a^{7}z^{7}+35a^{7}z^{5}-22a^{7}z^{3}+11a^{7}z-3a^{7}z^{-1}+2a^{6}z^{10}+a^{6}z^{8}-21a^{6}z^{6}+26a^{6}z^{4}-13a^{6}z^{2}+3a^{6}+4a^{5}z^{9}-13a^{5}z^{7}+11a^{5}z^{5}-7a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+11a^{4}z^{4}-3a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     2   -2 -4                   5 1   4 -6                 6 3     -3 -8               8 4       4 -10             7 6         -1 -12           9 8           1 -14         6 8             2 -16       4 8               -4 -18     2 6                 4 -20   1 4                   -3 -22   2                     2 -24 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ 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}} 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=-8} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ 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}} $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ 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}} $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ 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}^{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 r=-5} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ 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}^{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=-4} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ 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}^{8}} 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} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ 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=0} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ 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}} 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} ${\mathbb {Z} }_{2}$ 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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