L10a90

L10a89

L10a91

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a90 at Knotilus!

Link Presentations

[edit Notes on L10a90'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 X6,20,7,19 X16,7,17,8 X18,15,19,16 X8,17,1,18
Gauss code	{1, -5, 2, -6, 4, -7, 8, -10}, {5, -1, 3, -2, 6, -4, 9, -8, 10, -9, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)^{2}(t(2)-1)^{2}(t(1)t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+5q^{7/2}-{\frac {3}{q^{7/2}}}-8q^{5/2}+{\frac {5}{q^{5/2}}}+10q^{3/2}-{\frac {8}{q^{3/2}}}+q^{11/2}-11{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+2za^{-3}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-9z^{3}a^{-1}+6az-6za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-5az^{7}-6z^{7}a^{-1}-4z^{7}a^{-3}-a^{4}z^{6}+7a^{2}z^{6}+z^{6}a^{-2}-4z^{6}a^{-4}+13z^{6}+10a^{3}z^{5}+22az^{5}+18z^{5}a^{-1}+3z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}-a^{2}z^{4}+3z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}-9a^{3}z^{3}-22az^{3}-17z^{3}a^{-1}+4z^{3}a^{-5}-2a^{4}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-z^{2}+3a^{3}z+8az+6za^{-1}-za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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                 3 1   -2 6               5 2     3 4             5 3       -2 2           6 5         1 0         5 7           2 -2       3 4             -1 -4     2 5               3 -6   1 3                 -2 -8   2                   2 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

L10a89 L10a91