L10n104

L10n103

L10n105

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n104 at Knotilus!

Link Presentations

[edit Notes on L10n104's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X15,2,16,3 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code	{1, 4, -3, -10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {-4, 5, 10, 9, -8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)-t(3)t(4))(t(3)t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}-{\frac {1}{q^{13/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z^{-1}-3a^{7}z^{-3}-2a^{7}z-6a^{7}z^{-1}+a^{5}z^{3}+3a^{5}z^{-3}+6a^{5}z+9a^{5}z^{-1}-a^{3}z^{3}-a^{3}z^{-3}-4a^{3}z-4a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-7a^{9}z^{5}+15a^{9}z^{3}-a^{9}z^{-3}-12a^{9}z+5a^{9}z^{-1}+a^{8}z^{6}-7a^{8}z^{4}+14a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+a^{7}z^{7}-8a^{7}z^{5}+21a^{7}z^{3}-3a^{7}z^{-3}-23a^{7}z+12a^{7}z^{-1}+a^{6}z^{6}-8a^{6}z^{4}+20a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}-a^{5}z^{5}+7a^{5}z^{3}-3a^{5}z^{-3}-15a^{5}z+12a^{5}z^{-1}-a^{4}z^{4}+6a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{3}-a^{3}z^{-3}-4a^{3}z+5a^{3}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 1 1 -4             1   1 2 -6           1 4     3 -8             3     3 -10       1 3         2 -12         2         2 -14     1             1 -16 1                 1 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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