L11a511

L11a510

L11a512

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a511 at Knotilus!

Link Presentations

[edit Notes on L11a511's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X20,12,21,11 X18,10,19,9 X22,16,13,15 X12,20,7,19 X10,22,11,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -7, 3, -6}, {10, -2, 5, -8, 11, -4, 6, -3, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-3t(1)^{2}t(2)t(3)^{2}+5t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(3)-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)+3t(1)^{2}t(2)t(3)-5t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+11q^{5}-14q^{4}+17q^{3}-16q^{2}+15q-10+7q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-5a^{-2}-2z^{4}-3z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+5z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+6z^{8}a^{-6}-14z^{6}a^{-6}+17z^{4}a^{-6}-9z^{2}a^{-6}+2a^{-6}+4z^{9}a^{-5}-4z^{7}a^{-5}-2z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+z^{10}a^{-4}+10z^{8}a^{-4}-35z^{6}a^{-4}+53z^{4}a^{-4}-35z^{2}a^{-4}-a^{-4}z^{-2}+10a^{-4}+7z^{9}a^{-3}-13z^{7}a^{-3}+7z^{5}a^{-3}+5z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}+a^{2}z^{6}-27z^{6}a^{-2}-3a^{2}z^{4}+35z^{4}a^{-2}+3a^{2}z^{2}-29z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+12a^{-2}+3z^{9}a^{-1}+3az^{7}-z^{7}a^{-1}-8az^{5}-10z^{5}a^{-1}+5az^{3}+9z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+4z^{8}-8z^{6}+2z^{4}-2z^{2}-z^{-2}+4$ (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 17                       1 -1 15                     2   2 13                   4 1   -3 11                 7 2     5 9               8 5       -3 7             9 6         3 5           8 9           1 3         7 8             -1 1       4 9               5 -1     3 6                 -3 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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