L11n102

L11n101

L11n103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n102 at Knotilus!

Link Presentations

[edit Notes on L11n102's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -7, 9, -11, 2, -5, 6, -8, 3, -4, 5, -9, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-4t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-2a^{7}z^{3}-a^{7}z+a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-3a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-6z^{4}a^{10}+11z^{2}a^{10}-4a^{10}-3z^{7}a^{9}+3z^{5}a^{9}-z^{3}a^{9}+2za^{9}+a^{9}z^{-1}-5z^{8}a^{8}+13z^{6}a^{8}-24z^{4}a^{8}+25z^{2}a^{8}-9a^{8}-2z^{9}a^{7}-5z^{7}a^{7}+20z^{5}a^{7}-25z^{3}a^{7}+9za^{7}+a^{7}z^{-1}-10z^{8}a^{6}+23z^{6}a^{6}-24z^{4}a^{6}+12z^{2}a^{6}-4a^{6}-2z^{9}a^{5}-7z^{7}a^{5}+27z^{5}a^{5}-33z^{3}a^{5}+15za^{5}-2a^{5}z^{-1}-5z^{8}a^{4}+7z^{6}a^{4}-z^{4}a^{4}-3z^{2}a^{4}+2a^{4}-5z^{7}a^{3}+9z^{5}a^{3}-7z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               5 1   4 -4             7 4     -3 -6           7 3       4 -8         6 7         1 -10       7 7           0 -12     3 6             3 -14   3 7               -4 -16   3                 3 -18 3                   -3

Integral Khovanov Homology (db, data source)

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

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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