L11n240

L11n239

L11n241

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n240 at Knotilus!

Link Presentations

[edit Notes on L11n240's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X8,18,9,17
Gauss code	{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}-t(2)^{2}t(1)+2t(2)t(1)-t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+3q^{11/2}-4q^{9/2}+3q^{7/2}-3q^{5/2}+2q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-2z^{3}a^{-5}-5za^{-5}-4a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+10za^{-3}+6a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+2z^{7}a^{-7}-8z^{5}a^{-7}+8z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+z^{8}a^{-6}-z^{6}a^{-6}-8z^{4}a^{-6}+9z^{2}a^{-6}-3a^{-6}+4z^{7}a^{-5}-18z^{5}a^{-5}+24z^{3}a^{-5}-16za^{-5}+4a^{-5}z^{-1}+z^{8}a^{-4}-3z^{6}a^{-4}-2z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+3z^{7}a^{-3}-20z^{5}a^{-3}+39z^{3}a^{-3}-27za^{-3}+6a^{-3}z^{-1}+z^{8}a^{-2}-7z^{6}a^{-2}+10z^{4}a^{-2}-a^{-2}+az^{7}+2z^{7}a^{-1}-6az^{5}-16z^{5}a^{-1}+10az^{3}+33z^{3}a^{-1}-7az-22za^{-1}+2az^{-1}+5a^{-1}z^{-1}+z^{8}-6z^{6}+8z^{4}-2z^{2}-1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                   2 1   -1 10                 2 1     1 8             1 2 2       1 6             2 2         0 4         1 2 2           1 2       1 2 2             -1 0       1 3               2 -2   1 1                   0 -4                         0 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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