L11n152

L11n151

L11n153

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n152 at Knotilus!

Link Presentations

[edit Notes on L11n152's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X19,7,20,22 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 X2,18,3,17
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 $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$2q^{7/2}-4q^{5/2}+5q^{3/2}-7{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}-a^{3}z^{3}+3az^{3}-3z^{3}a^{-1}-a^{3}z+4az-6za^{-1}+2za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}-3a^{4}z^{4}+a^{4}z^{2}+2z^{2}a^{-4}-a^{-4}+3a^{3}z^{7}-11a^{3}z^{5}+z^{5}a^{-3}+9a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z-3za^{-3}+a^{-3}z^{-1}+3a^{2}z^{8}+z^{8}a^{-2}-10a^{2}z^{6}-3z^{6}a^{-2}+6a^{2}z^{4}+4z^{4}a^{-2}+3z^{2}a^{-2}-3a^{-2}+az^{9}+z^{9}a^{-1}+az^{7}-2z^{7}a^{-1}-14az^{5}-2z^{5}a^{-1}+18az^{3}+11z^{3}a^{-1}-9az-10za^{-1}+2az^{-1}+3a^{-1}z^{-1}+4z^{8}-14z^{6}+13z^{4}-3$ (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 χ 8                 2 -2 6               2   2 4             3 2   -1 2           4 2     2 0         3 4       1 -2       3 3         0 -4     2 4           2 -6   1 2             -1 -8   2               2 -10 1                 -1

Integral Khovanov Homology (db, data source)

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

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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