L11n245

L11n244

L11n246

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n245 at Knotilus!

Link Presentations

[edit Notes on L11n245's Link Presentations]

Planar diagram presentation	X12,1,13,2 X10,11,1,12 X14,5,15,6 X9,19,10,18 X17,3,18,2 X16,8,17,7 X3849 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code	{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(2u^{2}v-u^{2}+uv^{2}-uv+u-v^{2}+2v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {6}{q^{7/2}}}+6q^{5/2}-{\frac {9}{q^{5/2}}}-10q^{3/2}+{\frac {11}{q^{3/2}}}+11{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2az^{5}-z^{5}a^{-1}+2a^{3}z^{3}-5az^{3}-z^{3}a^{-1}+z^{3}a^{-3}+2a^{3}z-3az+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-5a^{2}z^{8}-4z^{8}a^{-2}-9z^{8}-4a^{3}z^{7}-3az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}-a^{4}z^{6}+10a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+22z^{6}+3a^{3}z^{5}+10az^{5}+16z^{5}a^{-1}+9z^{5}a^{-3}-6a^{4}z^{4}-14a^{2}z^{4}-5z^{4}a^{-2}+3z^{4}a^{-4}-16z^{4}-3a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-13z^{3}a^{-1}-8z^{3}a^{-3}+5a^{4}z^{2}+8a^{2}z^{2}-2z^{2}a^{-4}+5z^{2}+a^{5}z+4za^{-1}+3za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             6 2     4 2           5 4       -1 0         8 6         2 -2       5 7           2 -4     4 6             -2 -6   2 5               3 -8   4                 -4 -10 2                   2

Integral Khovanov Homology (db, data source)

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

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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