L10n73

L10n72

L10n74

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n73 at Knotilus!

Link Presentations

[edit Notes on L10n73's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X20,9,11,10 X18,12,19,11 X15,9,16,8 X10,19,5,20 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, 4, -7}, {5, -2, 10, 8, -6, 3, -8, -5, 7, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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