L10a62

L10a61

L10a63

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a62 at Knotilus!

Link Presentations

[edit Notes on L10a62's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{4}+3uv^{3}-3uv^{2}+3uv-2u+v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$8q^{9/2}-9q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-5q^{11/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+9z^{3}a^{-3}+7za^{-3}+a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-z^{2}a^{-8}+5z^{5}a^{-7}-3z^{3}a^{-7}+7z^{6}a^{-6}-10z^{4}a^{-6}+4z^{2}a^{-6}+7z^{7}a^{-5}-14z^{5}a^{-5}+9z^{3}a^{-5}-4za^{-5}+4z^{8}a^{-4}-3z^{6}a^{-4}-13z^{4}a^{-4}+11z^{2}a^{-4}-a^{-4}+z^{9}a^{-3}+8z^{7}a^{-3}-35z^{5}a^{-3}+36z^{3}a^{-3}-13za^{-3}+a^{-3}z^{-1}+6z^{8}a^{-2}-18z^{6}a^{-2}+8z^{4}a^{-2}+7z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-21z^{5}a^{-1}+9az^{3}+32z^{3}a^{-1}-7az-16za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}+z^{2}-3$ (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 16                     1 1 14                   2   -2 12                 3 1   2 10               5 2     -3 8             4 3       1 6           5 5         0 4         4 4           0 2       3 6             3 0     2 3               -1 -2   1 4                 3 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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