L11n38

L11n37

L11n39

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Link Presentations

[edit Notes on L11n38's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X5,12,6,13 X3849 X9,16,10,17 X13,19,14,18 X17,15,18,14 X15,10,16,11 X11,22,12,5 X19,2,20,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, -11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+3uv^{3}-uv^{2}-v^{3}+3v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+a^{9}z^{3}+5a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-9a^{7}z^{3}-10a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+8a^{5}z^{3}+7a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}-a^{12}+2a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-2a^{10}z^{4}+4a^{10}z^{2}-2a^{10}+3a^{9}z^{7}-12a^{9}z^{5}+18a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+11a^{8}z^{4}-4a^{8}z^{2}+a^{7}z^{9}+2a^{7}z^{7}-22a^{7}z^{5}+32a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+5a^{6}z^{8}-21a^{6}z^{6}+24a^{6}z^{4}-11a^{6}z^{2}+2a^{6}+a^{5}z^{9}-15a^{5}z^{5}+24a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-4a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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             3 2     -1 -8           3 2       1 -10       1 3 3         1 -12       4 3           1 -14     1 3             2 -16   1 3               -2 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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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