L10n26

L10n25

L10n27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n26 at Knotilus!

Link Presentations

[edit Notes on L10n26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,17,5,18 X18,14,19,13 X12,20,13,19 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 $u$, $v$, $w$, ...)	$-{\frac {uv^{3}-4uv^{2}+5uv-u-v^{3}+5v^{2}-4v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}-q^{7/2}+{\frac {4}{q^{7/2}}}+3q^{5/2}-{\frac {7}{q^{5/2}}}-5q^{3/2}+{\frac {7}{q^{3/2}}}+7{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+z^{3}a^{3}-za^{3}-a^{3}z^{-1}-z^{5}a-2z^{3}a-2za+2z^{3}a^{-1}+2za^{-1}-za^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}-2a^{3}z^{7}-5az^{7}-3z^{7}a^{-1}-a^{4}z^{6}-3a^{2}z^{6}-3z^{6}a^{-2}-5z^{6}+a^{3}z^{5}+6az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{4}z^{4}+4a^{2}z^{4}+7z^{4}a^{-2}+13z^{4}-3a^{5}z^{3}-3a^{3}z^{3}+2z^{3}a^{-1}+2z^{3}a^{-3}-a^{2}z^{2}-4z^{2}a^{-2}-5z^{2}+3a^{5}z+3a^{3}z-za^{-1}-za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               2   -2 4             3 1   2 2           4 2     -2 0         4 3       1 -2       4 5         1 -4     3 3           0 -6   1 4             3 -8 1 3               -2 -10 2                 2

Integral Khovanov Homology (db, data source)

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

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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