L11n99

L11n98

L11n100

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n99 at Knotilus!

Link Presentations

[edit Notes on L11n99's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}-uv^{2}-uv+2u+2v^{5}-v^{4}-v^{3}+v^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{15/2}-q^{13/2}+q^{11/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}-z^{5}a^{-3}+az^{3}-4z^{3}a^{-1}-5z^{3}a^{-3}+3az-2za^{-1}-6za^{-3}+2za^{-5}+za^{-7}+az^{-1}+a^{-1}z^{-1}-4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-2z^{8}a^{-2}-z^{8}a^{-4}-z^{8}-az^{7}+6z^{7}a^{-1}+8z^{7}a^{-3}-z^{7}a^{-7}+13z^{6}a^{-2}+8z^{6}a^{-4}-z^{6}a^{-6}-z^{6}a^{-8}+5z^{6}+6az^{5}-11z^{5}a^{-1}-22z^{5}a^{-3}+5z^{5}a^{-7}-25z^{4}a^{-2}-19z^{4}a^{-4}+6z^{4}a^{-6}+5z^{4}a^{-8}-5z^{4}-10az^{3}+10z^{3}a^{-1}+29z^{3}a^{-3}+4z^{3}a^{-5}-5z^{3}a^{-7}+19z^{2}a^{-2}+19z^{2}a^{-4}-7z^{2}a^{-6}-6z^{2}a^{-8}-z^{2}+5az-5za^{-1}-16za^{-3}-6za^{-5}-5a^{-2}-6a^{-4}+a^{-6}+2a^{-8}+1-az^{-1}+a^{-1}z^{-1}+4a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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 16                       1 -1 14                         0 12                   1 1   0 10               2 1       -1 8               1 1       0 6           2 3 1         0 4         1   1           2 2       1 3 1             1 0     1 1 1               1 -2     1                   1 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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