L10n82

L10n81

L10n83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n82 at Knotilus!

Link Presentations

[edit Notes on L10n82's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,20,12,13 X7,18,8,19 X9,16,10,17 X17,8,18,9 X13,10,14,11 X19,12,20,5 X2536 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{2}-uv^{2}w-uvw^{2}+uvw+uw^{3}-v^{3}-v^{2}w^{2}+v^{2}w+vw^{2}-vw}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-8}+q^{-6}+q^{-5}-2q^{-4}+3q^{-3}-2q^{-2}+q+3q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+a^{8}-a^{6}z^{2}-2a^{6}z^{-2}-3a^{6}+a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{4}-2a^{2}z^{2}+z^{2}+1$ (db)
Kauffman polynomial	$a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-22a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+a^{7}z^{7}-8a^{7}z^{5}+17a^{7}z^{3}-11a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-9a^{6}z^{6}+25a^{6}z^{4}-27a^{6}z^{2}-2a^{6}z^{-2}+13a^{6}+a^{5}z^{7}-8a^{5}z^{5}+19a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+3a^{4}z^{4}-6a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+2a^{3}z^{5}-2a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{2}+2az^{5}-4az^{3}+z^{4}-2z^{2}+1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                     1 1 1                   1   -1 -1                 2 1   1 -3               2 3     1 -5           1 2         1 -7           1 2         1 -9       1 2 2           -1 -11         2             2 -13     1                 1 -15 1                     1 -17 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $i=1$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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