L11a225

L11a224

L11a226

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a225 at Knotilus!

Link Presentations

[edit Notes on L11a225's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,16,21,15 X22,14,7,13 X14,22,15,21 X6,18,1,17
Gauss code	{1, -5, 6, -7, 2, -11}, {7, -1, 3, -4, 5, -6, 9, -10, 8, -2, 11, -3, 4, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{4}+8uv^{3}-11uv^{2}+8uv-2u-2v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+7q^{11/2}-11q^{9/2}+15q^{7/2}-18q^{5/2}+17q^{3/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+3z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{7}a^{-3}-4z^{5}a^{-3}-6z^{3}a^{-3}-4za^{-3}-2a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3az^{3}-6z^{3}a^{-1}+3az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-8z^{5}a^{-7}+4z^{3}a^{-7}+5z^{8}a^{-6}-14z^{6}a^{-6}+13z^{4}a^{-6}-9z^{2}a^{-6}+2a^{-6}+5z^{9}a^{-5}-13z^{7}a^{-5}+14z^{5}a^{-5}-12z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+2z^{10}a^{-4}+4z^{8}a^{-4}-24z^{6}a^{-4}+35z^{4}a^{-4}-22z^{2}a^{-4}+5a^{-4}+10z^{9}a^{-3}-28z^{7}a^{-3}+a^{3}z^{5}+32z^{5}a^{-3}-2a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+5z^{8}a^{-2}+3a^{2}z^{6}-22z^{6}a^{-2}-5a^{2}z^{4}+30z^{4}a^{-2}+a^{2}z^{2}-13z^{2}a^{-2}+3a^{-2}+5z^{9}a^{-1}+5az^{7}-7z^{7}a^{-1}-8az^{5}+z^{5}a^{-1}+4az^{3}+8z^{3}a^{-1}-4az-4za^{-1}+az^{-1}+6z^{8}-10z^{6}+6z^{4}-z^{2}-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     2   2 12                   5 1   -4 10                 6 2     4 8               9 5       -4 6             9 6         3 4           8 9           1 2         8 9             -1 0       4 9               5 -2     3 7                 -4 -4   1 5                   4 -6   2                     -2 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=2} $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}} ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}} $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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