L10n29

L10n28

L10n30

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n29 at Knotilus!

Link Presentations

[edit Notes on L10n29's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,20,16,5 X9,17,10,16 X19,9,20,8 X13,19,14,18 X17,13,18,12 X2536 X11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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