L10n65

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L10n66

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Link Presentations

[edit Notes on L10n65's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, -3, 6, -4, 8, -7, 3, -5, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-3uvw^{2}+3uvw-uv-2uw+u-vw^{3}+2vw^{2}+w^{3}-3w^{2}+3w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$3q^{-5}-5q^{-4}+q^{3}+7q^{-3}-2q^{2}-7q^{-2}+5q+8q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-2z^{4}-6z^{2}-3z^{-2}-6+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+5a^{3}z^{7}+7az^{7}+2z^{7}a^{-1}+7a^{4}z^{6}+10a^{2}z^{6}+z^{6}a^{-2}+4z^{6}+3a^{5}z^{5}-9a^{3}z^{5}-18az^{5}-6z^{5}a^{-1}-18a^{4}z^{4}-37a^{2}z^{4}-4z^{4}a^{-2}-23z^{4}+4a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}+6a^{6}z^{2}+24a^{4}z^{2}+40a^{2}z^{2}+6z^{2}a^{-2}+28z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 7                 1 1 5               1   -1 3             4 1   3 1           2 1     -1 -1         6 4       2 -3       4 5         1 -5     3 3           0 -7   2 4             2 -9 1 3               -2 -11 3                 3

Integral Khovanov Homology (db, data source)

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

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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