L10n45

L10n44

L10n46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n45 at Knotilus!

Link Presentations

[edit Notes on L10n45's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 X17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 4, -9, -7, 8, -10, -2, 9, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)t(1)-t(2)+1}{t(1)t(2)}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{5}+5az^{3}-z^{3}a^{-1}+6az+2az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z+a^{5}z^{-1}-a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+10a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+10a^{2}z^{4}-6a^{2}z^{2}+a^{2}+2az^{7}+z^{7}a^{-1}-12az^{5}-6z^{5}a^{-1}+20az^{3}+10z^{3}a^{-1}-11az-5za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-6z^{6}+10z^{4}-5z^{2}$ (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   \ -2 -1 0 1 2 3 4 χ 6             1 1 4               0 2         1 1   0 0     1 1       0 -2     1 1       0 -4 1 1 1         1 -6 1             1 -8 1             1

Integral Khovanov Homology (db, data source)

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

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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