L11a264

L11a263

L11a265

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a264 at Knotilus!

Link Presentations

[edit Notes on L11a264's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-5t(2)^{2}t(1)+7t(2)t(1)-3t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-10q^{11/2}+12q^{9/2}-15q^{7/2}+14q^{5/2}-12q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+6z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+2az-3za^{-1}+5za^{-3}-4za^{-5}+2za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-6z^{9}a^{-3}-3z^{9}a^{-5}-5z^{8}a^{-2}-7z^{8}a^{-4}-5z^{8}a^{-6}-3z^{8}-az^{7}+8z^{7}a^{-1}+12z^{7}a^{-3}-3z^{7}a^{-5}-6z^{7}a^{-7}+25z^{6}a^{-2}+21z^{6}a^{-4}+3z^{6}a^{-6}-5z^{6}a^{-8}+12z^{6}+4az^{5}+4z^{5}a^{-3}+18z^{5}a^{-5}+7z^{5}a^{-7}-3z^{5}a^{-9}-24z^{4}a^{-2}-12z^{4}a^{-4}+4z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}-14z^{4}-5az^{3}-8z^{3}a^{-1}-13z^{3}a^{-3}-18z^{3}a^{-5}-5z^{3}a^{-7}+3z^{3}a^{-9}+5z^{2}a^{-2}-3z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+5z^{2}+2az+2za^{-1}+6za^{-3}+9za^{-5}+2za^{-7}-za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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 6 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 6 2     4 10               7 5       -2 8             8 5         3 6           6 7           1 4         6 8             -2 2       4 7               3 0     2 5                 -3 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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