L11a280

L11a279

L11a281

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a280 at Knotilus!

Link Presentations

[edit Notes on L11a280's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X22,13,9,14 X14,6,15,5 X4,22,5,21 X18,8,19,7 X20,16,21,15 X16,20,17,19 X6,18,7,17
Gauss code	{1, -2, 3, -7, 6, -11, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -10, 11, -8, 10, -9, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{11/2}+3q^{9/2}-5q^{7/2}+8q^{5/2}-12q^{3/2}+13{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-2a^{3}z-za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+5az^{3}+4z^{3}a^{-1}+3az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-4z^{5}a^{-5}-3a^{5}z^{3}+4z^{3}a^{-5}+a^{5}z-za^{-5}+3z^{8}a^{-4}+5a^{4}z^{6}-13z^{6}a^{-4}-5a^{4}z^{4}+16z^{4}a^{-4}+2a^{4}z^{2}-5z^{2}a^{-4}+3z^{9}a^{-3}+6a^{3}z^{7}-10z^{7}a^{-3}-7a^{3}z^{5}+7z^{5}a^{-3}+4a^{3}z^{3}+z^{3}a^{-3}-a^{3}z-za^{-3}+z^{10}a^{-2}+5a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-24z^{6}a^{-2}-2a^{2}z^{4}+28z^{4}a^{-2}+2a^{2}z^{2}-9z^{2}a^{-2}+3az^{9}+6z^{9}a^{-1}+az^{7}-16z^{7}a^{-1}-11az^{5}+10z^{5}a^{-1}+9az^{3}-z^{3}a^{-1}-4az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+6z^{8}-20z^{6}+16z^{4}-5z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     2   -2 8                   3 1   2 6                 5 2     -3 4               7 3       4 2             6 5         -1 0           8 7           1 -2         6 8             2 -4       4 6               -2 -6     2 6                 4 -8   1 4                   -3 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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