L11n251

L11n250

L11n252

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n251 at Knotilus!

Link Presentations

[edit Notes on L11n251's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X8394 X11,17,12,16 X14,8,15,7 X6,16,7,15 X17,11,18,22 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}+t(1)^{2}+3t(2)t(1)+t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}-2q^{7/2}+{\frac {7}{q^{7/2}}}+5q^{5/2}-{\frac {11}{q^{5/2}}}-8q^{3/2}+{\frac {11}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+11{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-za^{-3}+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-2z^{3}a^{-1}+7az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-2a^{5}z^{3}+6a^{4}z^{6}-7a^{4}z^{4}+4a^{4}z^{2}+8a^{3}z^{7}-15a^{3}z^{5}+3z^{5}a^{-3}+16a^{3}z^{3}-6z^{3}a^{-3}-8a^{3}z+2za^{-3}+5a^{2}z^{8}+z^{8}a^{-2}-3a^{2}z^{6}+2z^{6}a^{-2}-6a^{2}z^{4}-9z^{4}a^{-2}+7a^{2}z^{2}+6z^{2}a^{-2}+az^{9}+z^{9}a^{-1}+11az^{7}+3z^{7}a^{-1}-30az^{5}-9z^{5}a^{-1}+29az^{3}+5z^{3}a^{-1}-14az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+6z^{8}-7z^{6}-7z^{4}+8z^{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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   2 2 6                 3   -3 4               5 2   3 2             6 3     -3 0           7 5       2 -2         6 8         2 -4       5 5           0 -6     2 6             4 -8   1 5               -4 -10   2                 2 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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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