L10n53

L10n52

L10n54

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Link Presentations

[edit Notes on L10n53's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,13,7,14 X9,15,10,14 X19,11,20,10 X5,16,6,17 X15,18,16,19 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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