L11n246

L11n245

L11n247

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n246 at Knotilus!

Link Presentations

[edit Notes on L11n246's Link Presentations]

Planar diagram presentation	X12,1,13,2 X3849 X5,14,6,15 X7,18,8,19 X9,21,10,20 X10,11,1,12 X13,6,14,7 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16
Gauss code	{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u+v)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+5uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{3/2}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-7a^{3}z-a^{3}z^{-1}+3az^{3}+3az-za^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-5a^{6}z^{8}-9a^{4}z^{8}-4a^{2}z^{8}-4a^{7}z^{7}-4a^{5}z^{7}-2a^{3}z^{7}-2az^{7}-a^{8}z^{6}+12a^{6}z^{6}+23a^{4}z^{6}+10a^{2}z^{6}+11a^{7}z^{5}+24a^{5}z^{5}+15a^{3}z^{5}+2az^{5}+2a^{8}z^{4}-3a^{6}z^{4}-15a^{4}z^{4}-14a^{2}z^{4}-4z^{4}-7a^{7}z^{3}-20a^{5}z^{3}-19a^{3}z^{3}-7az^{3}-z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+2a^{4}z^{2}+4a^{2}z^{2}+2z^{2}+a^{7}z+7a^{5}z+8a^{3}z+3az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   1 1 2                 3   -3 0               5 1   4 -2             6 4     -2 -4           6 4       2 -6         5 6         1 -8       5 6           -1 -10     3 6             3 -12   1 4               -3 -14   3                 3 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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