L11a364

L11a363

L11a365

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a364 at Knotilus!

Link Presentations

[edit Notes on L11a364's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-t(2)^{4}t(1)^{3}+3t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-2q^{15/2}+3q^{13/2}-5q^{11/2}+6q^{9/2}-7q^{7/2}+6q^{5/2}-6q^{3/2}+4{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+6z^{5}a^{-5}+11z^{3}a^{-5}+6za^{-5}-z^{9}a^{-3}-8z^{7}a^{-3}-23z^{5}a^{-3}-28z^{3}a^{-3}-12za^{-3}-a^{-3}z^{-1}+z^{7}a^{-1}+6z^{5}a^{-1}+11z^{3}a^{-1}+7za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-4z^{9}a^{-3}-2z^{9}a^{-5}+3z^{8}a^{-2}+3z^{8}a^{-4}-2z^{8}a^{-6}-2z^{8}-az^{7}+10z^{7}a^{-1}+22z^{7}a^{-3}+9z^{7}a^{-5}-2z^{7}a^{-7}+z^{6}a^{-2}-z^{6}a^{-4}+6z^{6}a^{-6}-2z^{6}a^{-8}+10z^{6}+5az^{5}-16z^{5}a^{-1}-45z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}-2z^{5}a^{-9}-6z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}a^{-6}+3z^{4}a^{-8}-z^{4}a^{-10}-13z^{4}-6az^{3}+14z^{3}a^{-1}+43z^{3}a^{-3}+15z^{3}a^{-5}-4z^{3}a^{-7}+4z^{3}a^{-9}+5z^{2}a^{-2}+5z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}a^{-10}+4z^{2}+az-7za^{-1}-14za^{-3}-5za^{-5}-za^{-9}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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                     1   1 14                   2 1   -1 12                 3 1     2 10               3 2       -1 8             4 3         1 6           3 4           1 4         3 3             0 2       2 4               2 0     1 2                 -1 -2   1 2                   1 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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