L10a89

L10a88

L10a90

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a89 at Knotilus!

Link Presentations

[edit Notes on L10a89's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X18,16,19,15 X16,7,17,8 X6,17,7,18 X8,20,1,19
Gauss code	{1, -5, 2, -6, 4, -9, 8, -10}, {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v-uv^{2}-u-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-4q^{11/2}+6q^{9/2}-8q^{7/2}+7q^{5/2}-7q^{3/2}+5{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+5za^{-3}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-7za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-2z^{2}a^{-8}+4z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+4z^{6}a^{-6}-2z^{4}a^{-6}-3z^{2}a^{-6}+4z^{7}a^{-5}-5z^{5}a^{-5}-2z^{3}a^{-5}+2za^{-5}+3z^{8}a^{-4}-5z^{6}a^{-4}-z^{4}a^{-4}+z^{2}a^{-4}+z^{9}a^{-3}+3z^{7}a^{-3}-18z^{5}a^{-3}+17z^{3}a^{-3}-4za^{-3}+5z^{8}a^{-2}-18z^{6}a^{-2}+15z^{4}a^{-2}-z^{2}a^{-2}+z^{9}a^{-1}+az^{7}-5az^{5}-14z^{5}a^{-1}+8az^{3}+23z^{3}a^{-1}-5az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{8}-9z^{6}+11z^{4}-3z^{2}-1$ (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 16                     1 1 14                   2   -2 12                 2 1   1 10               4 2     -2 8             4 2       2 6           3 4         1 4         4 4           0 2       3 5             2 0     1 2               -1 -2   1 3                 2 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a88 L10a90