L10n23

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Link Presentations

[edit Notes on L10n23's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,5,18,20 X13,19,14,18 X19,15,20,14 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$2q^{15/2}-2q^{13/2}+2q^{11/2}-3q^{9/2}+2q^{7/2}-3q^{5/2}+q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}+z^{5}a^{-7}+5z^{3}a^{-7}+6za^{-7}+2a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-12z^{3}a^{-5}-12za^{-5}-5a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+7za^{-3}+3a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{-10}+z^{3}a^{-9}-2za^{-9}+2z^{6}a^{-8}-8z^{4}a^{-8}+6z^{2}a^{-8}+3z^{7}a^{-7}-15z^{5}a^{-7}+21z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}-2z^{6}a^{-6}-7z^{4}a^{-6}+13z^{2}a^{-6}-5a^{-6}+4z^{7}a^{-5}-21z^{5}a^{-5}+32z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-4z^{6}a^{-4}+z^{4}a^{-4}+7z^{2}a^{-4}-5a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+12z^{3}a^{-3}-10za^{-3}+3a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 χ 16               2 -2 14             1 1 0 12           2 2   0 10         1 1 1   1 8       1 2       1 6     2 1         1 4   1 3           2 2                 0 0 1               1

Integral Khovanov Homology (db, data source)

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

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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