L10n61

L10n60

L10n62

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n61 at Knotilus!

Link Presentations

[edit Notes on L10n61's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 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 X8,15,9,16
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 $u$, $v$, $w$, ...)	${\frac {-2u^{3}v^{2}+2u^{3}v-u^{3}+2u^{2}v^{2}-2u^{2}v-2uv^{2}+2uv-v^{3}+2v^{2}-2v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-2z^{5}a^{3}-9z^{3}a^{3}-14za^{3}-7a^{3}z^{-1}+2z^{3}a+4za+2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+7a^{7}z^{3}-4a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}+2a^{6}z^{6}-17a^{6}z^{4}+20a^{6}z^{2}-8a^{6}+6a^{5}z^{7}-20a^{5}z^{5}+22a^{5}z^{3}-17a^{5}z+7a^{5}z^{-1}+a^{4}z^{8}+5a^{4}z^{6}-24a^{4}z^{4}+29a^{4}z^{2}-13a^{4}+4a^{3}z^{7}-12a^{3}z^{5}+18a^{3}z^{3}-16a^{3}z+7a^{3}z^{-1}+4a^{2}z^{6}-11a^{2}z^{4}+17a^{2}z^{2}-8a^{2}+az^{5}+3az^{3}-3az+2az^{-1}+3z^{2}-2$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                 2 -2 0               2   2 -2             4 3   -1 -4           3 1     2 -6         2 4       2 -8       4 3         1 -10     1 3           2 -12   1 3             -2 -14   1               1 -16 1                 -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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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