L11n92

L11n91

L11n93

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n92 at Knotilus!

Link Presentations

[edit Notes on L11n92's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X20,12,21,11 X22,18,5,17 X18,22,19,21 X10,20,11,19 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -8, -7, 4, -2, 11, 8, -9, 3, 5, -6, 7, -4, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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