L11n454

L11n453

L11n455

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n454 at Knotilus!

Link Presentations

[edit Notes on L11n454's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,16,3,15 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X17,1,18,4
Gauss code	{1, -4, -3, 11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {4, -5, -11, 9, -6, 8, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uvwx^{2}-2uvwx-uvx^{2}+uvx-uwx^{2}+2uwx+ux^{2}-2ux-2vw^{2}x+vw^{2}+2vwx-vw+w^{2}x-w^{2}-2wx+2w}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {9}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-3}-a^{9}z+a^{9}z^{-1}+a^{7}z^{5}+2a^{7}z^{3}-3a^{7}z^{-3}-a^{7}z-6a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+3a^{5}z^{-3}+7a^{5}z+9a^{5}z^{-1}-2a^{3}z^{3}-a^{3}z^{-3}-5a^{3}z-4a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-z^{2}a^{12}-3z^{7}a^{11}+11z^{5}a^{11}-10z^{3}a^{11}+3za^{11}-3z^{8}a^{10}+9z^{6}a^{10}-4z^{4}a^{10}-z^{2}a^{10}-z^{9}a^{9}-3z^{7}a^{9}+21z^{5}a^{9}-28z^{3}a^{9}+16za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-5z^{8}a^{8}+15z^{6}a^{8}-10z^{4}a^{8}-7z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{9}a^{7}-2z^{7}a^{7}+16z^{5}a^{7}-31z^{3}a^{7}+25za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-2z^{8}a^{6}+4z^{6}a^{6}-z^{4}a^{6}-15z^{2}a^{6}-6a^{6}z^{-2}+19a^{6}-2z^{7}a^{5}+6z^{5}a^{5}-16z^{3}a^{5}+19za^{5}-12a^{5}z^{-1}+3a^{5}z^{-3}-z^{6}a^{4}+2z^{4}a^{4}-8z^{2}a^{4}-3a^{4}z^{-2}+10a^{4}-3z^{3}a^{3}+7za^{3}-5a^{3}z^{-1}+a^{3}z^{-3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 2 2 0 -6               5     5 -8             3 4     1 -10           6 3       3 -12         3 5         2 -14       4 4           0 -16     2 5             3 -18   1 2               -1 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$

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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