L11n57

L11n56

L11n58

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n57 at Knotilus!

Link Presentations

[edit Notes on L11n57's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X19,5,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -4, 8, -9, 3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}-t(1)t(2)^{4}-3t(2)^{4}+2t(2)^{3}+2t(1)t(2)^{2}-3t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {2}{q^{9/2}}}-q^{7/2}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{5/2}}}+q^{3/2}-{\frac {4}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z-2a^{3}z^{-1}+3za^{-3}+a^{-3}z^{-1}+az^{5}-z^{5}a^{-1}+4az^{3}-6z^{3}a^{-1}+6az+3az^{-1}-8za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-4a^{2}z^{8}-2z^{8}-a^{5}z^{7}+a^{3}z^{7}+2az^{7}-z^{7}a^{-1}-z^{7}a^{-3}+9a^{4}z^{6}+19a^{2}z^{6}-z^{6}a^{-4}+11z^{6}+5a^{5}z^{5}+12a^{3}z^{5}+10az^{5}+9z^{5}a^{-1}+6z^{5}a^{-3}-10a^{4}z^{4}-23a^{2}z^{4}+4z^{4}a^{-2}+5z^{4}a^{-4}-14z^{4}-8a^{5}z^{3}-20a^{3}z^{3}-22az^{3}-18z^{3}a^{-1}-8z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}-7z^{2}a^{-2}-5z^{2}a^{-4}+5z^{2}+5a^{5}z+10a^{3}z+14az+13za^{-1}+4za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                         0 6                 1 1 1   1 4               2 1       -1 2             2 1 1       2 0           3 4 1         0 -2         2 2 1           1 -4       2 3 1             0 -6     2 2                 0 -8   1 3                   2 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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