L11n197

L11n196

L11n198

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Link Presentations

[edit Notes on L11n197's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,19,6,18 X21,16,22,17 X15,20,16,21 X19,14,20,15 X2738 X4,11,5,12 X17,1,18,6
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -7, 6, -11, 5, -8, 7, -6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}-u^{2}+uv^{4}-uv^{3}-uv^{2}-uv+u-v^{4}+v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-q^{5/2}+2q^{3/2}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+a^{5}z^{3}+3a^{5}z+3a^{5}z^{-1}-a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+3az-2za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}-a^{7}z^{7}+6a^{3}z^{7}+4az^{7}-z^{7}a^{-1}-a^{8}z^{6}-a^{6}z^{6}+7a^{4}z^{6}+18a^{2}z^{6}+11z^{6}+5a^{7}z^{5}+a^{5}z^{5}-9a^{3}z^{5}+5z^{5}a^{-1}+5a^{8}z^{4}+6a^{6}z^{4}-11a^{4}z^{4}-28a^{2}z^{4}-16z^{4}-6a^{7}z^{3}-a^{5}z^{3}+6a^{3}z^{3}-5az^{3}-6z^{3}a^{-1}-6a^{8}z^{2}-7a^{6}z^{2}+4a^{4}z^{2}+12a^{2}z^{2}+7z^{2}+3a^{7}z+4a^{5}z-a^{3}z+2za^{-1}+a^{8}+3a^{6}+3a^{4}-a^{7}z^{-1}-3a^{5}z^{-1}-2a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     1   -1 2                   1 1   0 0               1 2 1     0 -2             1 2 1       0 -4           1 1 2         2 -6         1 3 1           1 -8       1   1             2 -10       1 1               0 -12   1 1                   0 -14                         0 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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