L11a300

L11a299

L11a301

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a300 at Knotilus!

Link Presentations

[edit Notes on L11a300's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+9t(2)^{2}t(1)^{2}-7t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-7t(2)^{2}t(1)+9t(2)t(1)-3t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$15q^{9/2}-18q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-4q^{15/2}+8q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+7z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}+2az-4za^{-1}+6za^{-3}-3za^{-5}+za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-2z^{3}a^{-9}+8z^{6}a^{-8}-8z^{4}a^{-8}+2z^{2}a^{-8}+10z^{7}a^{-7}-13z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+8z^{8}a^{-6}-6z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+4z^{9}a^{-5}+7z^{7}a^{-5}-30z^{5}a^{-5}+23z^{3}a^{-5}-7za^{-5}+a^{-5}z^{-1}+z^{10}a^{-4}+12z^{8}a^{-4}-33z^{6}a^{-4}+21z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+7z^{9}a^{-3}-11z^{7}a^{-3}-11z^{5}a^{-3}+19z^{3}a^{-3}-6za^{-3}+a^{-3}z^{-1}+z^{10}a^{-2}+7z^{8}a^{-2}-31z^{6}a^{-2}+32z^{4}a^{-2}-9z^{2}a^{-2}+3z^{9}a^{-1}+az^{7}-7z^{7}a^{-1}-4az^{5}-2z^{5}a^{-1}+5az^{3}+10z^{3}a^{-1}-2az-3za^{-1}+3z^{8}-12z^{6}+15z^{4}-6z^{2}$ (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 18                       1 -1 16                     3   3 14                   5 1   -4 12                 7 3     4 10               9 6       -3 8             9 6         3 6           7 9           2 4         7 9             -2 2       4 8               4 0     2 6                 -4 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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