L10n57

L10n56

L10n58

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n57 at Knotilus!

Link Presentations

[edit Notes on L10n57's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,5,9,6 X3,15,4,14 X15,5,16,4 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X6,19,7,20
Gauss code	{1, -8, -3, 4, 2, -10, -5, 9}, {8, -1, -6, 7, -9, 3, -4, 5, -7, 6, 10, -2}

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	${\frac {1}{q^{9/2}}}-q^{7/2}-{\frac {1}{q^{7/2}}}+q^{5/2}-{\frac {1}{q^{3/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}-3a^{3}z-za^{-3}+az^{5}+5az^{3}+5az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{7}-az^{7}-a^{4}z^{6}-a^{2}z^{6}-z^{6}a^{-2}-z^{6}+6a^{3}z^{5}+7az^{5}-z^{5}a^{-3}+5a^{4}z^{4}+6a^{2}z^{4}+5z^{4}a^{-2}+6z^{4}-10a^{3}z^{3}-14az^{3}+4z^{3}a^{-3}-5a^{4}z^{2}-7a^{2}z^{2}-5z^{2}a^{-2}-7z^{2}+6a^{3}z+10az+2za^{-1}-2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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 8                   1 1 6                     0 4             1 1 1   -1 2           1 1       0 0         1 3 1       1 -2       1 1 2         2 -4       1             1 -6   1 1 1             1 -8                     0 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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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