L11n428

L11n427

L11n429

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n428 at Knotilus!

Link Presentations

[edit Notes on L11n428's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X15,1,16,6 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in 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 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} , ...)	${\frac {(w-1)(uv-1)^{2}}{uv{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{4}+q^{3}+q+1+q^{-2}+q^{-4}$ (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^4 z^{-2} +2 a^4-a^2 z^4-z^4 a^{-2} -6 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} -7 a^2-2 a^{-2} +z^6+6 z^4+10 z^2+ z^{-2} +7} (db)
Kauffman 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^4 z^4-4 a^4 z^2-a^4 z^{-2} +5 a^4+z^7 a^{-3} -6 z^5 a^{-3} +a^3 z^3+9 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +z^8 a^{-2} -2 a^2 z^6-7 z^6 a^{-2} +13 a^2 z^4+15 z^4 a^{-2} -24 a^2 z^2-12 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a^{-2} +z^7 a^{-1} -a z^5-7 z^5 a^{-1} +5 a z^3+13 z^3 a^{-1} -7 a z-6 z a^{-1} +2 a z^{-1} +z^8-9 z^6+27 z^4-32 z^2- z^{-2} +13} (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 $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                     0 5             1 1 1   1 3             1       1 1         2 1 1       2 -1       1 3 1         1 -3     1 1 1           1 -5   1 3 1             1 -7     1               1 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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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