L11n63

L11n62

L11n64

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n63 at Knotilus!

Link Presentations

[edit Notes on L11n63's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}} $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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