L11n139

L11n138

L11n140

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n139 at Knotilus!

Link Presentations

[edit Notes on L11n139'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 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, 7, 3, -2, 5, 9, -10, 8, -7, -4, 2, -11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-u^{2}v+uv^{2}-uv+u-v+1}{uv}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-{\sqrt {q}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-5z^{3}a^{-3}-6za^{-3}-2a^{-3}z^{-1}+az+za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+15z^{4}a^{-6}-11z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}-7z^{7}a^{-5}+15z^{5}a^{-5}-12z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-14z^{6}a^{-4}+29z^{4}a^{-4}-21z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-7z^{7}a^{-3}+15z^{5}a^{-3}-14z^{3}a^{-3}+8za^{-3}-2a^{-3}z^{-1}+z^{8}a^{-2}-7z^{6}a^{-2}+14z^{4}a^{-2}-10z^{2}a^{-2}+3a^{-2}+az^{3}-z^{3}a^{-1}-3az+za^{-1}+az^{-1}-1$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 14                   1 -1 12                     0 10               1 1   0 8           1 1       0 6             1       1 4       1 2 1         0 2     1               1 0     2 1             1 -2 1                   1 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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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