L11n227

L11n226

L11n228

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n227 at Knotilus!

Link Presentations

[edit Notes on L11n227's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X17,21,18,20 X6,14,7,13 X14,8,15,7 X8,16,1,15 X19,4,20,5 X5,18,6,19 X21,17,22,16
Gauss code	{1, -4, 2, 9, -10, -6, 7, -8}, {4, -1, 3, -2, 6, -7, 8, 11, -5, 10, -9, 5, -11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v-uv^{2}-uv-u-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$5q^{9/2}-6q^{7/2}+6q^{5/2}-{\frac {1}{q^{5/2}}}-7q^{3/2}+{\frac {2}{q^{3/2}}}+q^{13/2}-3q^{11/2}+5{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-8z^{3}a^{-1}+8z^{3}a^{-3}-4z^{3}a^{-5}+3az-7za^{-1}+6za^{-3}-3za^{-5}+za^{-7}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-3z^{8}a^{-4}-2z^{8}-az^{7}-2z^{7}a^{-3}-3z^{7}a^{-5}+19z^{6}a^{-2}+9z^{6}a^{-4}-z^{6}a^{-6}+9z^{6}+5az^{5}+14z^{5}a^{-1}+17z^{5}a^{-3}+8z^{5}a^{-5}-17z^{4}a^{-2}-7z^{4}a^{-4}-z^{4}a^{-6}-11z^{4}-8az^{3}-22z^{3}a^{-1}-18z^{3}a^{-3}-7z^{3}a^{-5}-3z^{3}a^{-7}+3z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}+5az+10za^{-1}+6za^{-3}+2za^{-5}+za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (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 14                   1 -1 12                 2   2 10               3 1   -2 8             3 2     1 6           3 3       0 4         4 3         1 2       3 5           2 0     1 2             -1 -2   1 3               2 -4   1                 -1 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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