L11n147

L11n146

L11n148

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Link Presentations

[edit Notes on L11n147's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code	{1, 11, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 9, -10, -7, 8, -11, -2, 4, -9, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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