L10n60

L10n59

L10n61

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n60 at Knotilus!

Link Presentations

[edit Notes on L10n60's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X8,15,9,16
Gauss code	{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, -7, 8, 10, -2, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)^{3}-2t(2)^{2}t(1)^{2}+2t(2)t(1)^{2}+2t(2)^{2}t(1)-2t(2)t(1)-t(2)^{3}}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+2q^{11/2}-2q^{9/2}+3q^{7/2}-2q^{5/2}+2q^{3/2}-2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+9za^{-3}+5a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-7z^{3}a^{-1}+4az-13za^{-1}+4az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+2z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-2a^{-6}+z^{7}a^{-5}-2z^{5}a^{-5}-2z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}+3z^{6}a^{-4}-13z^{4}a^{-4}+18z^{2}a^{-4}-9a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-8za^{-3}+5a^{-3}z^{-1}+2z^{6}a^{-2}-14z^{4}a^{-2}+28z^{2}a^{-2}-14a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-9z^{5}a^{-1}+15az^{3}+23z^{3}a^{-1}-13az-21za^{-1}+4az^{-1}+8a^{-1}z^{-1}+z^{6}-8z^{4}+15z^{2}-8$ (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 5 6 χ 14                     1 1 12                   1   -1 10                 1 1   0 8               2 1     -1 6           1 1 1       -1 4           2 2         0 2       1 2 1           0 0         3             3 -2     1                 1 -4 1                     1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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