L11n114

L11n113

L11n115

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n114 at Knotilus!

Link Presentations

[edit Notes on L11n114's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, 9, -7, -2, 11, 8, -9, 7, -8, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (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	$-z^{9}a^{-1}-z^{9}a^{-3}-7z^{8}a^{-2}-5z^{8}a^{-4}-2z^{8}-az^{7}-3z^{7}a^{-1}-10z^{7}a^{-3}-8z^{7}a^{-5}+23z^{6}a^{-2}+10z^{6}a^{-4}-5z^{6}a^{-6}+8z^{6}+5az^{5}+26z^{5}a^{-1}+47z^{5}a^{-3}+25z^{5}a^{-5}-z^{5}a^{-7}-16z^{4}a^{-2}+2z^{4}a^{-4}+9z^{4}a^{-6}-9z^{4}-9az^{3}-41z^{3}a^{-1}-58z^{3}a^{-3}-31z^{3}a^{-5}-5z^{3}a^{-7}-8z^{2}a^{-4}-9z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+7az+24za^{-1}+31za^{-3}+18za^{-5}+4za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -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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