L11n144

L11n143

L11n145

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n144 at Knotilus!

Link Presentations

[edit Notes on L11n144's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 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 $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {5}{q^{7/2}}}+q^{5/2}-{\frac {7}{q^{5/2}}}-2q^{3/2}+{\frac {6}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-az^{5}-3az^{3}+z^{3}a^{-1}-4az-2az^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-3a^{2}z^{8}-z^{8}-3a^{5}z^{7}+2a^{3}z^{7}+5az^{7}-2a^{6}z^{6}+5a^{4}z^{6}+12a^{2}z^{6}+5z^{6}-a^{7}z^{5}+10a^{5}z^{5}-a^{3}z^{5}-14az^{5}-2z^{5}a^{-1}+5a^{6}z^{4}-3a^{4}z^{4}-22a^{2}z^{4}-z^{4}a^{-2}-15z^{4}+3a^{7}z^{3}-11a^{5}z^{3}-4a^{3}z^{3}+14az^{3}+4z^{3}a^{-1}-a^{6}z^{2}-2a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}+17z^{2}-a^{7}z+6a^{5}z+2a^{3}z-7az-2za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 1   1 2               3 1   -2 0             3 1     2 -2           4 4       0 -4         3 2         1 -6       2 4           2 -8     2 3             -1 -10     2               2 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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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