L10n63

L10n62

L10n64

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n63 at Knotilus!

Link Presentations

[edit Notes on L10n63's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X3948 X17,2,18,3 X5,14,6,15 X11,6,12,7 X9,18,10,19 X15,11,16,20 X10,13,1,14 X19,5,20,4
Gauss code	{1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(uv+1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-7q^{3/2}+9{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{8}-3z^{8}-6a^{3}z^{7}-12az^{7}-6z^{7}a^{-1}-3a^{4}z^{6}-a^{2}z^{6}-4z^{6}a^{-2}-2z^{6}+13a^{3}z^{5}+26az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+6a^{2}z^{4}+7z^{4}a^{-2}+13z^{4}-6a^{5}z^{3}-18a^{3}z^{3}-18az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+6a^{5}z+9a^{3}z+4az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               3   -3 4             4 1   3 2           5 3     -2 0         6 4       2 -2       5 6         1 -4     4 5           -1 -6   2 5             3 -8 1 4               -3 -10 3                 3

Integral Khovanov Homology (db, data source)

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

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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