L10a22

L10a21

L10a23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a22 at Knotilus!

Link Presentations

[edit Notes on L10a22's Link Presentations]

Planar diagram presentation	X6172 X14,9,15,10 X4,19,1,20 X12,6,13,5 X10,4,11,3 X20,12,5,11 X2,14,3,13 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code	{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-12q^{11/2}+15q^{9/2}-17q^{7/2}+14q^{5/2}-12q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-5z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+6z^{3}a^{-3}+3za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-10z^{4}a^{-8}+5z^{2}a^{-8}-2a^{-8}+11z^{7}a^{-7}-15z^{5}a^{-7}+8z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+7z^{8}a^{-6}+z^{6}a^{-6}-19z^{4}a^{-6}+14z^{2}a^{-6}-5a^{-6}+2z^{9}a^{-5}+16z^{7}a^{-5}-41z^{5}a^{-5}+26z^{3}a^{-5}-7za^{-5}+2a^{-5}z^{-1}+12z^{8}a^{-4}-19z^{6}a^{-4}-4z^{4}a^{-4}+11z^{2}a^{-4}-3a^{-4}+2z^{9}a^{-3}+9z^{7}a^{-3}-33z^{5}a^{-3}+26z^{3}a^{-3}-6za^{-3}+5z^{8}a^{-2}-10z^{6}a^{-2}+2z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+9z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                     1 -1 16                   3   3 14                 6 1   -5 12               6 3     3 10             9 6       -3 8           8 6         2 6         6 9           3 4       6 8             -2 2     3 8               5 0   1 4                 -3 -2   3                   3 -4 1                     -1

Integral Khovanov Homology (db, data source)

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

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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