L11n131

L11n130

L11n132

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n131 at Knotilus!

Link Presentations

[edit Notes on L11n131's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21
Gauss code	{1, 4, -3, 10, -5, -6}, {6, -1, 7, 3, -2, 5, 8, -11, 9, -7, -4, 2, -10, -8, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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