L11a26

L11a25

L11a27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a26 at Knotilus!

Link Presentations

[edit Notes on L11a26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,5,15 X16,22,17,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 7, -8, 4, -5, 6, -4, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(2)^{3}-5t(2)^{3}-8t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-8t(2)-5t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-11q^{9/2}+14q^{7/2}-{\frac {1}{q^{7/2}}}-16q^{5/2}+{\frac {2}{q^{5/2}}}+16q^{3/2}-{\frac {6}{q^{3/2}}}+q^{15/2}-3q^{13/2}+7q^{11/2}-14{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-2z^{3}a^{-5}+a^{-5}z^{-1}+z^{5}a^{-3}-z^{3}a^{-3}+a^{3}z-3za^{-3}+a^{3}z^{-1}-2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}-2az+za^{-1}-az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-8z^{5}a^{-7}+5z^{3}a^{-7}-za^{-7}+5z^{8}a^{-6}-14z^{6}a^{-6}+13z^{4}a^{-6}-9z^{2}a^{-6}+3a^{-6}+4z^{9}a^{-5}-7z^{7}a^{-5}+z^{5}a^{-5}-z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+9z^{8}a^{-4}-34z^{6}a^{-4}+46z^{4}a^{-4}-30z^{2}a^{-4}+7a^{-4}+7z^{9}a^{-3}-16z^{7}a^{-3}+a^{3}z^{5}+20z^{5}a^{-3}-3a^{3}z^{3}-18z^{3}a^{-3}+3a^{3}z+10za^{-3}-a^{3}z^{-1}-2a^{-3}z^{-1}+z^{10}a^{-2}+7z^{8}a^{-2}+2a^{2}z^{6}-21z^{6}a^{-2}-3a^{2}z^{4}+28z^{4}a^{-2}-16z^{2}a^{-2}+a^{2}+4a^{-2}+3z^{9}a^{-1}+3az^{7}-3z^{7}a^{-1}-3az^{5}+7z^{5}a^{-1}-az^{3}-10z^{3}a^{-1}+2az+6za^{-1}-az^{-1}-a^{-1}z^{-1}+3z^{8}-5z^{4}+3z^{2}$ (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                     2   2 12                   5 1   -4 10                 6 2     4 8               8 5       -3 6             8 6         2 4           8 8           0 2         6 8             -2 0       4 9               5 -2     2 5                 -3 -4     4                   4 -6 1 2                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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