L10n108

L10n107

L10n109

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n108 at Knotilus!

Link Presentations

[edit Notes on L10n108's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{2}x^{2}-uvw^{2}x-uw^{2}x^{2}+uwx^{2}+vw-v-x+1}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-z^{5}a^{7}-5z^{3}a^{7}-7za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+z^{7}a^{5}+6z^{5}a^{5}+12z^{3}a^{5}+13za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}-z^{5}a^{3}-5z^{3}a^{3}-7za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}$ (db)
Kauffman polynomial	$a^{11}z+a^{10}z^{2}+3a^{9}z^{3}-a^{9}z^{-3}-8a^{9}z+5a^{9}z^{-1}+2a^{8}z^{6}-9a^{8}z^{4}+13a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+3a^{7}z^{7}-16a^{7}z^{5}+28a^{7}z^{3}-3a^{7}z^{-3}-25a^{7}z+12a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}-9a^{6}z^{4}+23a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+4a^{5}z^{7}-22a^{5}z^{5}+37a^{5}z^{3}-3a^{5}z^{-3}-27a^{5}z+12a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+11a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 0                 1 1 -2                   0 -4             3 1   2 -6         1 1 2     2 -8         4 1       3 -10     2 1 2         3 -12     4 2           2 -14 1   1             2 -16 2 2               0 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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