L10n103

L10n102

L10n104

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n103 at Knotilus!

Link Presentations

[edit Notes on L10n103's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, {10, -2, -5, 8, 4, -3, 6, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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