L10n106

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L10n105

L10n107

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n106 at Knotilus!

Link Presentations

[edit Notes on L10n106's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code	{1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -6, 8}, {-4, -5, 10, 9, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)-1)(t(4)-1)(t(1)t(3)t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+a^{7}z^{-3}+z^{5}a^{5}+4z^{3}a^{5}+3za^{5}-2a^{5}z^{-1}-3a^{5}z^{-3}-z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}-2za^{3}+4a^{3}z^{-1}+3a^{3}z^{-3}+z^{5}a+3z^{3}a-2az^{-1}-az^{-3}$ (db)
Kauffman polynomial	$-z^{2}a^{8}-4z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}-a^{7}z^{-3}-2z^{6}a^{6}+4z^{4}a^{6}-3z^{2}a^{6}+3a^{6}z^{-2}-4a^{6}-4z^{7}a^{5}+16z^{5}a^{5}-24z^{3}a^{5}+11za^{5}+3a^{5}z^{-1}-3a^{5}z^{-3}-2z^{8}a^{4}+4z^{6}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+6a^{4}z^{-2}-7a^{4}-7z^{7}a^{3}+28z^{5}a^{3}-32z^{3}a^{3}+11za^{3}+3a^{3}z^{-1}-3a^{3}z^{-3}-2z^{8}a^{2}+5z^{6}a^{2}+2z^{4}a^{2}-3z^{2}a^{2}+3a^{2}z^{-2}-4a^{2}-3z^{7}a+12z^{5}a-12z^{3}a+3za+2az^{-1}-az^{-3}-z^{6}+3z^{4}-z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 4                 1 -1 2               2   2 0             1 1   0 -2           5 2     3 -4       1 2 3       2 -6       6 3         3 -8   1 2 5           4 -10   3 2             1 -12   3               3 -14 1                 -1

Integral Khovanov Homology (db, data source)

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

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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