L10n70

L10n69

L10n71

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n70 at Knotilus!

Link Presentations

[edit Notes on L10n70's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10}, {-7, 2, 4, -5, 9, 7, -8, 6, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+2q^{-7}-2q^{-6}+3q^{-5}-2q^{-4}+3q^{-3}+q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{2}a^{4}-2a^{4}z^{-2}-a^{4}+z^{2}a^{2}+a^{2}z^{-2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+3a^{10}z^{2}-a^{10}+2a^{9}z^{7}-9a^{9}z^{5}+9a^{9}z^{3}-2a^{9}z+a^{8}z^{8}-3a^{8}z^{6}-2a^{8}z^{4}+6a^{8}z^{2}-a^{8}+3a^{7}z^{7}-14a^{7}z^{5}+17a^{7}z^{3}-6a^{7}z+a^{6}z^{8}-4a^{6}z^{6}+3a^{6}z^{4}+a^{6}z^{2}+a^{6}z^{-2}-a^{6}+a^{5}z^{7}-5a^{5}z^{5}+8a^{5}z^{3}-2a^{5}z-2a^{5}z^{-1}+a^{4}z^{4}-a^{4}z^{2}+2a^{4}z^{-2}-2a^{4}+2a^{3}z-2a^{3}z^{-1}+a^{2}z^{2}+a^{2}z^{-2}-2a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                 1 1 -3               1 2 1 -5             2   1 3 -7           1 2     1 -9         3 2       1 -11       1 3 1       1 -13     1 2 1         0 -15   1 1             0 -17   1               -1 -19 1                 1

Integral Khovanov Homology (db, data source)

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

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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