L11n238

L11n237

L11n239

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n238 at Knotilus!

Link Presentations

[edit Notes on L11n238's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,16,8,17 X5,1,6,10 X3746 X9,5,10,4 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X17,8,18,9
Gauss code	{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, {10, -1, -6, 9, -8, 2, -11, 6, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+6u^{2}v^{2}-6u^{2}v+2u^{2}+2uv^{3}-6uv^{2}+6uv-2u-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-2z^{5}a^{-1}-4a^{3}z^{3}+11az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-8a^{3}z+12az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-5a^{3}z^{-1}+6az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-5a^{2}z^{8}-3z^{8}a^{-2}-8z^{8}-7a^{3}z^{7}-14az^{7}-10z^{7}a^{-1}-3z^{7}a^{-3}-3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}-z^{6}a^{-4}+12z^{6}+17a^{3}z^{5}+44az^{5}+36z^{5}a^{-1}+9z^{5}a^{-3}+9z^{4}a^{-2}+3z^{4}a^{-4}+6z^{4}-6a^{5}z^{3}-29a^{3}z^{3}-51az^{3}-37z^{3}a^{-1}-9z^{3}a^{-3}-a^{4}z^{2}-3a^{2}z^{2}-11z^{2}a^{-2}-3z^{2}a^{-4}-10z^{2}+7a^{5}z+21a^{3}z+29az+19za^{-1}+4za^{-3}+a^{4}+a^{2}+3a^{-2}+a^{-4}+3-2a^{5}z^{-1}-5a^{3}z^{-1}-6az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -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           6 4       -2 0         7 6         1 -2       6 7           1 -4     4 6             -2 -6   2 6               4 -8 1 4                 -3 -10 3                   3

Integral Khovanov Homology (db, data source)

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

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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