L10a69

L10a68

L10a70

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a69 at Knotilus!

Link Presentations

[edit Notes on L10a69's Link Presentations]

Planar diagram presentation	X8192 X16,9,17,10 X6718 X20,17,7,18 X10,4,11,3 X14,6,15,5 X4,12,5,11 X18,14,19,13 X12,20,13,19 X2,16,3,15
Gauss code	{1, -10, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 8, -6, 10, -2, 4, -8, 9, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(v^{2}-3v+1\right)(uv-u+1)(uv-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-7q^{9/2}+11q^{7/2}-14q^{5/2}+15q^{3/2}-15{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+6z^{3}a^{-3}+6za^{-3}+2a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+az-4za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+za^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+5z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-3za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-6z^{6}a^{-4}+3z^{4}a^{-4}-2z^{2}a^{-4}+2z^{9}a^{-3}+8z^{7}a^{-3}+a^{3}z^{5}-23z^{5}a^{-3}-a^{3}z^{3}+23z^{3}a^{-3}-11za^{-3}+2a^{-3}z^{-1}+11z^{8}a^{-2}+4a^{2}z^{6}-20z^{6}a^{-2}-6a^{2}z^{4}+14z^{4}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}+a^{-2}+2z^{9}a^{-1}+7az^{7}+10z^{7}a^{-1}-13az^{5}-28z^{5}a^{-1}+7az^{3}+24z^{3}a^{-1}-2az-9za^{-1}+az^{-1}+2a^{-1}z^{-1}+6z^{8}-7z^{6}$ (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                 5 1   4 8               6 2     -4 6             8 5       3 4           8 7         -1 2         7 7           0 0       5 9             4 -2     3 6               -3 -4   1 5                 4 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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