L10a33

L10a32

L10a34

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Link Presentations

[edit Notes on L10a33's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X20,13,5,14 X8,18,9,17 X14,19,15,20 X10,16,11,15 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -8, 10, -2, 5, -7, 8, -3, 6, -4, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{4}-4uv^{3}+4uv^{2}-4uv+u+v^{5}-4v^{4}+4v^{3}-4v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {2}{q^{9/2}}}+3q^{7/2}-{\frac {5}{q^{7/2}}}-6q^{5/2}+{\frac {6}{q^{5/2}}}+8q^{3/2}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-9{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-z^{3}a^{-3}-8a^{3}z-4a^{3}z^{-1}-za^{-3}+2az^{5}+z^{5}a^{-1}+7az^{3}+z^{3}a^{-1}+8az+3az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-2a^{3}z^{7}-8az^{7}-7z^{7}a^{-1}+8a^{4}z^{6}+18a^{2}z^{6}-8z^{6}a^{-2}+2z^{6}+5a^{5}z^{5}+21a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-6z^{5}a^{-3}-8a^{4}z^{4}-10a^{2}z^{4}+12z^{4}a^{-2}-3z^{4}a^{-4}+13z^{4}-9a^{5}z^{3}-33a^{3}z^{3}-33az^{3}-3z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-4a^{2}z^{2}-6z^{2}a^{-2}-9z^{2}+7a^{5}z+17a^{3}z+14az+2za^{-1}-2za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^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 3 4 χ 10                     1 1 8                   2   -2 6                 4 1   3 4               4 2     -2 2             5 4       1 0           6 5         -1 -2         3 4           -1 -4       3 6             3 -6     2 3               -1 -8   1 4                 3 -10   1                   -1 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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