L10a64

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L10a65

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a64 at Knotilus!

Link Presentations

[edit Notes on L10a64's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X16,14,17,13 X14,5,15,6 X4,15,5,16 X18,12,19,11 X12,18,13,17 X6,20,1,19
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 8, -9, 5, -6, 7, -5, 9, -8, 10, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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