L10n56

L10n55

L10n57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n56 at Knotilus!

Link Presentations

[edit Notes on L10n56's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X20,5,9,6 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X8,9,1,10 X4,19,5,20 X2,18,3,17
Gauss code	{1, -10, 5, -9, 3, 7, -4, -8}, {8, -1, 2, -5, -6, 4, -7, 6, 10, -2, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(u+1)^{2}(v-1)}{u^{3/2}{\sqrt {v}}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{5}(-z)-a^{3}z+az^{5}+5az^{3}-z^{3}a^{-1}+5az+az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}-a^{3}z^{7}-2az^{7}-z^{7}a^{-1}+6a^{2}z^{6}+6z^{6}-a^{5}z^{5}+6a^{3}z^{5}+13az^{5}+6z^{5}a^{-1}-a^{6}z^{4}-8a^{2}z^{4}-9z^{4}+4a^{5}z^{3}-8a^{3}z^{3}-22az^{3}-10z^{3}a^{-1}+3a^{6}z^{2}+a^{4}z^{2}+a^{2}z^{2}+3z^{2}-2a^{5}z+2a^{3}z+10az+6za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                     0 2               1 1   0 0           2 1       1 -2         1 2 1       0 -4       1 1 1         1 -6       1             1 -8   1 1 1             1 -10                     0 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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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