L11n33

L11n32

L11n34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n33 at Knotilus!

Link Presentations

[edit Notes on L11n33's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$2q^{9/2}+{\frac {1}{q^{9/2}}}-3q^{7/2}-{\frac {2}{q^{7/2}}}+3q^{5/2}+{\frac {1}{q^{5/2}}}-2q^{3/2}-{\frac {1}{q^{3/2}}}-q^{11/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+5az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}-2a^{3}z+6az-8za^{-1}+5za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-5z^{5}a^{-5}+7z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+2z^{8}a^{-4}+a^{4}z^{6}-10z^{6}a^{-4}-4a^{4}z^{4}+13z^{4}a^{-4}+2a^{4}z^{2}-7z^{2}a^{-4}+a^{-4}+z^{9}a^{-3}+2a^{3}z^{7}-2z^{7}a^{-3}-10a^{3}z^{5}-11z^{5}a^{-3}+11a^{3}z^{3}+24z^{3}a^{-3}-5a^{3}z-16za^{-3}+3a^{-3}z^{-1}+a^{2}z^{8}+4z^{8}a^{-2}-5a^{2}z^{6}-23z^{6}a^{-2}+3a^{2}z^{4}+34z^{4}a^{-2}-17z^{2}a^{-2}+a^{2}+3a^{-2}+z^{9}a^{-1}+3az^{7}-2z^{7}a^{-1}-21az^{5}-17z^{5}a^{-1}+34az^{3}+40z^{3}a^{-1}-17az-24za^{-1}+2az^{-1}+4a^{-1}z^{-1}+3z^{8}-19z^{6}+28z^{4}-12z^{2}+2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     1   -1 8                   2 1   1 6               1 2 1     0 4             1 2 2       -1 2           2 3 2         1 0         2 4 2           0 -2       1 2 3             2 -4     1 2 1               0 -6   1 1 1                 1 -8   1                     1 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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