L10n14

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n14 at Knotilus!

Link Presentations

[edit Notes on L10n14's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 9, -8, 6, -7, 4, -9, 8, 10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-8za^{3}-3a^{3}z^{-1}+z^{5}a+6z^{3}a+9za+4az^{-1}-z^{3}a^{-1}-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-7a^{6}z^{4}+4a^{6}z^{2}-a^{6}+a^{5}z^{7}-3a^{5}z^{5}+a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+2a^{4}z^{6}-9a^{4}z^{4}+10a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-13a^{3}z^{5}+24a^{3}z^{3}-15a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+6a^{2}z^{4}+5a^{2}z^{2}-3a^{2}+2az^{7}+z^{7}a^{-1}-15az^{5}-6z^{5}a^{-1}+30az^{3}+10z^{3}a^{-1}-19az-7za^{-1}+4az^{-1}+2a^{-1}z^{-1}+z^{8}-6z^{6}+8z^{4}-z^{2}-2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                     1 1 4                       0 2                 1 1   0 0             3 1       2 -2           1 2 1       0 -4         2 2 1         1 -6       1 1             0 -8     1 2 1             0 -10   1 1                 0 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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