L11n137

L11n136

L11n138

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Link Presentations

[edit Notes on L11n137'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 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15
Gauss code	{1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, -8, 11, -9, -7, 4, -2, 10, 8, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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