L11n190

L11n189

L11n191

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Link Presentations

[edit Notes on L11n190's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, 5, -11, -7, 8, -6, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-u^{2}v^{3}+4u^{2}v^{2}-3u^{2}v+u^{2}+4uv^{3}-7uv^{2}+4uv+v^{4}-3v^{3}+4v^{2}-v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {6}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {10}{q^{13/2}}}-{\frac {12}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-4a^{7}z^{3}-6a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+a^{5}z^{3}+a^{5}z-2a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+9z^{5}a^{11}-8z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-3z^{8}a^{10}+4z^{6}a^{10}+8z^{4}a^{10}-11z^{2}a^{10}+3a^{10}-z^{9}a^{9}-7z^{7}a^{9}+26z^{5}a^{9}-24z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-6z^{8}a^{8}+8z^{6}a^{8}+7z^{4}a^{8}-9z^{2}a^{8}+3a^{8}-z^{9}a^{7}-7z^{7}a^{7}+19z^{5}a^{7}-16z^{3}a^{7}+7za^{7}-2a^{7}z^{-1}-3z^{8}a^{6}+2z^{6}a^{6}-2z^{4}a^{6}+2z^{2}a^{6}-3z^{7}a^{5}+2z^{5}a^{5}-3z^{3}a^{5}-z^{6}a^{4}-4z^{4}a^{4}+3z^{2}a^{4}-3z^{3}a^{3}+2za^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 4 1 -3 -6               4 1   3 -8             6 4     -2 -10           6 4       2 -12         4 6         2 -14       5 6           -1 -16     2 5             3 -18   1 4               -3 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$

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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