L10n64

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Link Presentations

[edit Notes on L10n64's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X3948 X17,2,18,3 X14,6,15,5 X6,12,7,11 X9,18,10,19 X20,15,11,16 X10,13,1,14 X4,19,5,20
Gauss code	{1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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