L11n53

L11n52

L11n54

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n53 at Knotilus!

Link Presentations

[edit Notes on L11n53's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X13,20,14,21 X17,5,18,22 X21,19,22,18 X19,14,20,15 X15,8,16,9 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 9, -4, -2, 11, -3, -5, 8, -9, 4, -6, 7, -8, 5, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+{\frac {1}{q^{7/2}}}+4q^{5/2}-{\frac {3}{q^{5/2}}}-5q^{3/2}+{\frac {4}{q^{3/2}}}+5{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+2az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-a^{3}z+3az-3za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-4}-3z^{4}a^{-4}+z^{2}a^{-4}+3z^{7}a^{-3}-11z^{5}a^{-3}+2a^{3}z^{3}+8z^{3}a^{-3}-2a^{3}z-2za^{-3}+a^{2}z^{8}+3z^{8}a^{-2}-5a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+8z^{4}a^{-2}-3a^{2}z^{2}-z^{2}a^{-2}+az^{9}+z^{9}a^{-1}-3az^{7}-az^{5}-12z^{5}a^{-1}+10az^{3}+16z^{3}a^{-1}-6az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+4z^{8}-17z^{6}+20z^{4}-5z^{2}-1$ (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 χ 10                 1 -1 8               2   2 6             2 1   -1 4           3 2     1 2       1 3 2       0 0       4 3         1 -2     2 4           2 -4   1 2             -1 -6   2               2 -8 1                 -1

Integral Khovanov Homology (db, data source)

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

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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