L11a303

L11a302

L11a304

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Link Presentations

[edit Notes on L11a303's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,8,19,7 X20,14,21,13 X14,22,15,21 X22,18,9,17 X8,20,1,19 X6,9,7,10 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10, 5, -9}, {10, -1, 2, -3, 6, -7, 11, -4, 8, -5, 9, -6, 7, -8}

A Braid Representative
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} , 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} , 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} , ...)	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 -\frac{u^3 v^5-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+5 u^2 v^4-8 u^2 v^3+7 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-8 u v^2+5 u v-u-v^3+2 v^2-2 v+1}{u^{3/2} v^{5/2}}} (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-12q^{3/2}+16{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (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^3 z^7+5 a^3 z^5+9 a^3 z^3+6 a^3 z+2 a^3 z^{-1} -a z^9-7 a z^7+z^7 a^{-1} -19 a z^5+5 z^5 a^{-1} -24 a z^3+9 z^3 a^{-1} -14 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} } (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 -2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-10 a z^9-5 z^9 a^{-1} -6 a^4 z^8-5 a^2 z^8-5 z^8 a^{-2} -4 z^8-5 a^5 z^7+6 a^3 z^7+26 a z^7+12 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+14 a^2 z^6+13 z^6 a^{-2} -z^6 a^{-4} +17 z^6-a^7 z^5+7 a^5 z^5-7 a^3 z^5-38 a z^5-15 z^5 a^{-1} +8 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-15 a^2 z^4-11 z^4 a^{-2} +3 z^4 a^{-4} -20 z^4+2 a^7 z^3-3 a^5 z^3+9 a^3 z^3+35 a z^3+16 z^3 a^{-1} -5 z^3 a^{-3} -2 a^6 z^2+9 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-4} +14 z^2-a^7 z+a^5 z-7 a^3 z-16 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums 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 \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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   5 1   -4 4                 7 2     5 2               9 5       -4 0             10 7         3 -2           9 10           1 -4         8 9             -1 -6       4 9               5 -8     3 8                 -5 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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 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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