L11n241

L11n240

L11n242

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n241 at Knotilus!

Link Presentations

[edit Notes on L11n241's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 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 $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-t(2)t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$7q^{9/2}-9q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}+2q^{13/2}-5q^{11/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-4z^{3}a^{-5}-6za^{-5}-4a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+12za^{-3}+6a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-10za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{2}a^{-8}-a^{-8}+z^{5}a^{-7}+5z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+5z^{6}a^{-6}-9z^{4}a^{-6}+9z^{2}a^{-6}-3a^{-6}+8z^{7}a^{-5}-25z^{5}a^{-5}+31z^{3}a^{-5}-18za^{-5}+4a^{-5}z^{-1}+5z^{8}a^{-4}-10z^{6}a^{-4}-2z^{4}a^{-4}+8z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+10z^{7}a^{-3}-47z^{5}a^{-3}+58z^{3}a^{-3}-31za^{-3}+6a^{-3}z^{-1}+7z^{8}a^{-2}-23z^{6}a^{-2}+16z^{4}a^{-2}-a^{-2}+z^{9}a^{-1}+az^{7}+3z^{7}a^{-1}-5az^{5}-26z^{5}a^{-1}+9az^{3}+41z^{3}a^{-1}-7az-24za^{-1}+2az^{-1}+5a^{-1}z^{-1}+2z^{8}-8z^{6}+9z^{4}-2z^{2}-1$ (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 χ 14                   2 -2 12                 3   3 10               4 2   -2 8             5 3     2 6           4 4       0 4         5 5         0 2       4 6           2 0     1 3             -2 -2   1 4               3 -4   1                 -1 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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