L11n225

L11n224

L11n226

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n225 at Knotilus!

Link Presentations

[edit Notes on L11n225's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,12,4,13 X16,9,17,10 X11,20,12,21 X22,15,9,16 X5,14,6,15 X13,4,14,5 X6,20,7,19 X18,8,19,7 X8,18,1,17 X21,3,22,2
Gauss code	{1, 11, -2, 7, -6, -8, 9, -10}, {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}+2t(2)^{2}t(1)-t(2)t(1)+2t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+a^{5}z^{5}+a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+2a^{3}z^{5}+7a^{3}z^{3}+8a^{3}z+3a^{3}z^{-1}-2az^{3}-5az-2az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+7z^{4}a^{8}-3z^{2}a^{8}-5z^{7}a^{7}+8z^{5}a^{7}-7z^{3}a^{7}+2za^{7}-3z^{8}a^{6}+3z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-a^{6}-z^{9}a^{5}-2z^{7}a^{5}+2z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-4z^{8}a^{4}+10z^{6}a^{4}-14z^{4}a^{4}+12z^{2}a^{4}-3a^{4}-z^{9}a^{3}+3z^{7}a^{3}-12z^{5}a^{3}+24z^{3}a^{3}-15za^{3}+3a^{3}z^{-1}-z^{8}a^{2}+2z^{6}a^{2}-2z^{4}a^{2}+4z^{2}a^{2}-3a^{2}-3z^{5}a+10z^{3}a-9za+2az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   2 2 0                 1   -1 -2               5 2   3 -4             5 3     -2 -6           5 3       2 -8         4 5         1 -10       4 5           -1 -12     2 4             2 -14   1 4               -3 -16   2                 2 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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