L11n23

L11n22

L11n24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n23 at Knotilus!

Link Presentations

[edit Notes on L11n23's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, 11, -2, -3, 6, -10, 4, -6, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-4v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {4}{q^{5/2}}}+7q^{3/2}-{\frac {7}{q^{3/2}}}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{-5}z^{-1}-a^{3}z^{3}-a^{3}z+4za^{-3}+3a^{-3}z^{-1}+az^{5}+3az^{3}-4z^{3}a^{-1}+5az-8za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-4az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}+9a^{2}z^{6}+12z^{6}a^{-2}+22z^{6}+11a^{3}z^{5}+26az^{5}+22z^{5}a^{-1}+7z^{5}a^{-3}+3a^{4}z^{4}-5a^{2}z^{4}-23z^{4}a^{-2}-3z^{4}a^{-4}-28z^{4}-10a^{3}z^{3}-34az^{3}-40z^{3}a^{-1}-17z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+15z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}+3a^{3}z+15az+24za^{-1}+14za^{-3}+2za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 χ 10                   1 1 8                 2   -2 6               4 1   3 4             3 2     -1 2           5 4       1 0         5 5         0 -2       2 3           -1 -4     2 5             3 -6   1 2               -1 -8   2                 2 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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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