L10n59

L10n58

L10n60

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n59 at Knotilus!

Link Presentations

[edit Notes on L10n59's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X15,9,16,8
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 $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(2)-t(1))(t(1)t(2)-1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+6az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-3a^{3}z+11az-11za^{-1}+3za^{-3}-2a^{3}z^{-1}+7az^{-1}-7a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{7}-az^{7}-z^{7}a^{-1}-z^{7}a^{-3}-a^{4}z^{6}-2a^{2}z^{6}-2z^{6}a^{-2}-z^{6}a^{-4}-2z^{6}+5a^{3}z^{5}+7az^{5}+7z^{5}a^{-1}+5z^{5}a^{-3}+5a^{4}z^{4}+12a^{2}z^{4}+12z^{4}a^{-2}+5z^{4}a^{-4}+14z^{4}-6a^{3}z^{3}-14az^{3}-14z^{3}a^{-1}-6z^{3}a^{-3}-6a^{4}z^{2}-18a^{2}z^{2}-18z^{2}a^{-2}-6z^{2}a^{-4}-24z^{2}+4a^{3}z+14az+14za^{-1}+4za^{-3}+2a^{4}+8a^{2}+8a^{-2}+2a^{-4}+13-2a^{3}z^{-1}-7az^{-1}-7a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                       0 6                 1 1   0 4             1 1       0 2           1   1       2 0         1 4 1         2 -2       1   1           2 -4       1 1             0 -6   1 1                 0 -8                       0 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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