L10n46

L10n45

L10n47

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Link Presentations

[edit Notes on L10n46's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X13,20,14,7 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X19,14,20,15 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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