L10a59

L10a58

L10a60

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a59 at Knotilus!

Link Presentations

[edit Notes on L10a59's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v-u^{2}+uv^{4}-5uv^{3}+7uv^{2}-5uv+u-v^{4}+3v^{3}-3v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-9q^{5/2}+11q^{3/2}-12{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+2az^{3}-2z^{3}a^{-3}-a^{3}z+3az-3za^{-1}-za^{-3}+za^{-5}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-6az^{7}-8z^{7}a^{-1}-5z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}-4z^{6}a^{-2}-5z^{6}a^{-4}+8z^{6}+10a^{3}z^{5}+23az^{5}+20z^{5}a^{-1}+4z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+a^{2}z^{4}+15z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+6z^{4}-10a^{3}z^{3}-23az^{3}-17z^{3}a^{-1}-z^{3}a^{-3}+3z^{3}a^{-5}-2a^{4}z^{2}-4a^{2}z^{2}-12z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-9z^{2}+3a^{3}z+11az+10za^{-1}+za^{-3}-za^{-5}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 4 1   -3 6               5 2     3 4             6 4       -2 2           6 5         1 0         5 7           2 -2       4 5             -1 -4     2 6               4 -6   1 3                 -2 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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