L10n105

L10n104

L10n106

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n105 at Knotilus!

Link Presentations

[edit Notes on L10n105's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,16,3,15 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X17,1,18,4
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 $u$, $v$, $w$, ...)	${\frac {t(1)t(4)t(3)^{2}-t(4)^{2}t(3)-t(1)t(2)t(3)-t(1)t(4)t(3)+t(1)t(2)t(4)t(3)-t(2)t(4)t(3)+t(4)t(3)+t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {4}{q^{5/2}}}-2q^{3/2}+{\frac {2}{q^{3/2}}}+{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$2a^{5}z^{-1}+a^{5}z^{-3}-4za^{3}-7a^{3}z^{-1}-3a^{3}z^{-3}+2z^{3}a+6za+8az^{-1}+3az^{-3}-2za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-a^{5}z^{-3}-8a^{5}z+5a^{5}z^{-1}+a^{4}z^{6}-4a^{4}z^{4}+10a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{7}-5a^{3}z^{5}+15a^{3}z^{3}-3a^{3}z^{-3}-19a^{3}z+12a^{3}z^{-1}+2a^{2}z^{6}-8a^{2}z^{4}+20a^{2}z^{2}+6a^{2}z^{-2}-19a^{2}+az^{7}-5az^{5}+15az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-19az-8za^{-1}+12az^{-1}+5a^{-1}z^{-1}+z^{6}-4z^{4}+10z^{2}+3z^{-2}-10$ (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   \ -4 -3 -2 -1 0 1 2 χ 4             2 2 2           1 2 1 0         3     3 -2       1 3     2 -4     3 1       2 -6     3         3 -8 2 1           1 -10 2             2

Integral Khovanov Homology (db, data source)

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

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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