L11a104

L11a103

L11a105

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a104 at Knotilus!

Link Presentations

[edit Notes on L11a104's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X20,11,21,12 X18,6,19,5 X22,20,5,19 X10,21,11,22 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code	{1, -2, 3, -8}, {5, -1, 10, -11, 2, -7, 4, -3, 9, -10, 11, -9, 8, -5, 6, -4, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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 v} , $w$, ...)	${\frac {uv^{5}-6uv^{4}+12uv^{3}-12uv^{2}+4uv+4v^{4}-12v^{3}+12v^{2}-6v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-5q^{7/2}+10q^{5/2}-15q^{3/2}+20{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	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 a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+9 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-9 a^3 z+8 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} } (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-12az^{9}-7z^{9}a^{-1}-7a^{4}z^{8}-12a^{2}z^{8}-9z^{8}a^{-2}-14z^{8}-6a^{5}z^{7}-4a^{3}z^{7}+15az^{7}+8z^{7}a^{-1}-5z^{7}a^{-3}-3a^{6}z^{6}+8a^{4}z^{6}+27a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+38z^{6}-a^{7}z^{5}+10a^{5}z^{5}+24a^{3}z^{5}+11az^{5}+8z^{5}a^{-1}+10z^{5}a^{-3}+4a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-19z^{4}+2a^{7}z^{3}-10a^{5}z^{3}-32a^{3}z^{3}-24az^{3}-7z^{3}a^{-1}-3z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}-4a^{2}z^{2}-z^{2}a^{-2}-4z^{2}-a^{7}z+7a^{5}z+18a^{3}z+13az+2za^{-1}-za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 9 4     5 2               11 6       -5 0             12 9         3 -2           11 12           1 -4         8 11             -3 -6       5 11               6 -8     3 8                 -5 -10   1 6                   5 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ 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 r=-6} ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ 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}^{6}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{3}$ $r=-3$ 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}^{8}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ 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}^{4}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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