L10n25

L10n24

L10n26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n25 at Knotilus!

Link Presentations

[edit Notes on L10n25's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X17,5,18,20 X13,18,14,19 X19,12,20,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8, -7, 4, -5, 3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}-uv^{2}-2uv+u+v^{5}-2v^{4}-v^{3}+v^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-q^{7/2}-{\frac {2}{q^{7/2}}}+q^{5/2}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-2z^{3}a^{3}-7za^{3}-4a^{3}z^{-1}+z^{5}a+5z^{3}a+6za+3az^{-1}-za^{-1}-a^{-1}z^{-1}-za^{-3}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-3a^{3}z^{7}-2az^{7}+5a^{4}z^{6}+5a^{2}z^{6}-z^{6}a^{-2}-z^{6}+6a^{5}z^{5}+18a^{3}z^{5}+13az^{5}-z^{5}a^{-3}-5a^{4}z^{4}-3a^{2}z^{4}+5z^{4}a^{-2}+7z^{4}-11a^{5}z^{3}-30a^{3}z^{3}-22az^{3}+z^{3}a^{-1}+4z^{3}a^{-3}-a^{4}z^{2}-5a^{2}z^{2}-5z^{2}a^{-2}-9z^{2}+8a^{5}z+18a^{3}z+13az+za^{-1}-2za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                       0 4               1 1 1   -1 2             1 1       0 0           1 2 1       0 -2         1 2 2         1 -4       1 1             0 -6     1 1 1             1 -8   1 2                 1 -10                       0 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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