L10n77

L10n76

L10n78

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n77 at Knotilus!

Link Presentations

[edit Notes on L10n77's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w^{3}-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-12}+q^{-8}+q^{-6}+q^{-4}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-a^{10}z^{6}-7a^{10}z^{4}-15a^{10}z^{2}-2a^{10}z^{-2}-11a^{10}+a^{8}z^{8}+8a^{8}z^{6}+21a^{8}z^{4}+21a^{8}z^{2}+a^{8}z^{-2}+8a^{8}$ (db)
Kauffman polynomial	$a^{16}+a^{12}z^{4}-5a^{12}z^{2}-a^{12}z^{-2}+5a^{12}+a^{11}z^{7}-7a^{11}z^{5}+15a^{11}z^{3}-11a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+22a^{10}z^{4}-26a^{10}z^{2}-2a^{10}z^{-2}+13a^{10}+a^{9}z^{7}-7a^{9}z^{5}+15a^{9}z^{3}-11a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-8a^{8}z^{6}+21a^{8}z^{4}-21a^{8}z^{2}-a^{8}z^{-2}+8a^{8}$ (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 χ -7                 1 1 -9                 1 1 -11             1     1 -13         1         1 -15         2 1       1 -17     1             1 -19     1 1           0 -21 1 1               0 -23 2 1               1 -25 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-9$ $i=-7$ $i=-5$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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