L10n30

L10n29

L10n31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n30 at Knotilus!

Link Presentations

[edit Notes on L10n30's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X15,5,16,20 X7,17,8,16 X17,12,18,13 X9,14,10,15 X13,18,14,19 X19,9,20,8 X2536 X11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-3t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-3t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{5/2}}}+q^{3/2}-{\frac {7}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+4z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+z^{5}a+3z^{3}a+2za+az^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{8}+a^{8}-z^{5}a^{7}-5z^{3}a^{7}+4za^{7}-a^{7}z^{-1}-5z^{6}a^{6}+11z^{4}a^{6}-13z^{2}a^{6}+4a^{6}-6z^{7}a^{5}+18z^{5}a^{5}-24z^{3}a^{5}+16za^{5}-4a^{5}z^{-1}-2z^{8}a^{4}-3z^{6}a^{4}+21z^{4}a^{4}-21z^{2}a^{4}+7a^{4}-9z^{7}a^{3}+29z^{5}a^{3}-28z^{3}a^{3}+15za^{3}-4a^{3}z^{-1}-2z^{8}a^{2}+z^{6}a^{2}+13z^{4}a^{2}-14z^{2}a^{2}+4a^{2}-3z^{7}a+10z^{5}a-9z^{3}a+3za-az^{-1}-z^{6}+3z^{4}-3z^{2}+1$ (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 χ 4                 1 -1 2               2   2 0             3 1   -2 -2           4 2     2 -4         4 4       0 -6       4 3         1 -8     2 4           2 -10   3 4             -1 -12   3               3 -14 2                 -2

Integral Khovanov Homology (db, data source)

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

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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