L11n148

L11n147

L11n149

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n148 at Knotilus!

Link Presentations

[edit Notes on L11n148's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X19,7,20,22 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 X17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, -9, 10, 7, -8, -11, 2, -4, 9, -10, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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