L11n426

L11n425

L11n427

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n426 at Knotilus!

Link Presentations

[edit Notes on L11n426's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X14,6,15,5 X3,10,4,11 X4,14,5,13 X17,2,18,3 X9,19,10,18 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code	{1, 6, -4, -5, 3, -11}, {2, -1, -7, 4, 9, -8}, {5, -3, 10, -2, -6, 7, 11, -10, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-t(2))^{2}(t(3)-1)}{t(1)t(2){\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-5}-q^{-4}-q^{3}+q^{-3}+q^{2}+q^{-1}+2$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{4}z^{2}+a^{4}z^{-2}+2a^{4}-a^{2}z^{4}-5a^{2}z^{2}-2a^{2}z^{-2}-z^{2}a^{-2}-7a^{2}-2a^{-2}+z^{4}+5z^{2}+z^{-2}+7$ (db)
Kauffman polynomial	$a^{4}z^{8}-7a^{4}z^{6}+15a^{4}z^{4}-12a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+a^{3}z^{9}-7a^{3}z^{7}+14a^{3}z^{5}-7a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z-2za^{-3}+2a^{3}z^{-1}+2a^{2}z^{8}-15a^{2}z^{6}+35a^{2}z^{4}+z^{4}a^{-2}-32a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4a^{-2}+az^{9}-7az^{7}+13az^{5}-z^{5}a^{-1}-3az^{3}+5z^{3}a^{-1}-7az-6za^{-1}+2az^{-1}+z^{8}-8z^{6}+21z^{4}-24z^{2}-z^{-2}+13$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   1 -1 5                     0 3             1 1 1   1 1             2       2 -1         1 1 3       3 -3       1 2           1 -5       1 2           1 -7   1 1               0 -9                     0 -11 1                   1

Integral Khovanov Homology (db, data source)

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

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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