L11n172

L11n171

L11n173

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Link Presentations

[edit Notes on L11n172's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X9,18,10,19 X20,11,21,12 X4,18,5,17 X19,3,20,2
Gauss code	{1, 11, -2, -10, 5, -6}, {3, -1, -8, 2, 9, -3, 6, -5, 4, -7, 10, 8, -11, -9, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-3u^{2}v+2u^{2}-5uv^{2}+9uv-5u+2v^{2}-3v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {10}{q^{7/2}}}-{\frac {11}{q^{5/2}}}-2q^{3/2}+{\frac {9}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+za^{5}-a^{5}z^{-1}-z^{5}a^{3}-z^{3}a^{3}+2a^{3}z^{-1}+3z^{3}a+3za-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+6a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-12a^{6}z^{6}+10a^{6}z^{4}-6a^{6}z^{2}+2a^{6}+2a^{5}z^{9}-a^{5}z^{7}-8a^{5}z^{5}+5a^{5}z^{3}-a^{5}z^{-1}+8a^{4}z^{8}-25a^{4}z^{6}+29a^{4}z^{4}-18a^{4}z^{2}+5a^{4}+2a^{3}z^{9}-a^{3}z^{7}-4a^{3}z^{5}+4a^{3}z^{3}+2a^{3}z-2a^{3}z^{-1}+4a^{2}z^{8}-11a^{2}z^{6}+18a^{2}z^{4}-11a^{2}z^{2}+3a^{2}+3az^{7}-5az^{5}+8az^{3}+3z^{3}a^{-1}-3az-4za^{-1}+a^{-1}z^{-1}+z^{6}+2z^{4}-z^{2}-1$ (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 4                   2 2 2                 3 1 -2 0               5 1   4 -2             5 4     -1 -4           6 4       2 -6         4 5         1 -8       4 6           -2 -10     2 4             2 -12   1 4               -3 -14   2                 2 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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