L11n439

L11n438

L11n440

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Link Presentations

[edit Notes on L11n439's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X22,19,13,20 X20,15,21,16 X16,21,17,22 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, 9, -10, -11, 3, 8, -9, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uvw-uvx^{3}+uvx^{2}-uvx+uv-uwx^{2}+uwx+ux^{3}+vw+vx^{2}-vx+wx^{3}-wx^{2}+wx-w-x^{3}}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-q^{11/2}+q^{9/2}-5q^{7/2}+2q^{5/2}-5q^{3/2}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{5}-2z^{5}a^{-1}-a^{3}z^{3}+7az^{3}-11z^{3}a^{-1}+3z^{3}a^{-3}-3a^{3}z+15az-22za^{-1}+11za^{-3}-za^{-5}-3a^{3}z^{-1}+14az^{-1}-22a^{-1}z^{-1}+14a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{3}z^{-3}+5az^{-3}-9a^{-1}z^{-3}+7a^{-3}z^{-3}-2a^{-5}z^{-3}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-a^{3}z^{7}-az^{7}-5z^{7}a^{-1}-6z^{7}a^{-3}-z^{7}a^{-5}-a^{4}z^{6}-2a^{2}z^{6}-3z^{6}a^{-2}+2z^{6}a^{-4}-6z^{6}+5a^{3}z^{5}+9az^{5}+28z^{5}a^{-1}+30z^{5}a^{-3}+6z^{5}a^{-5}+5a^{4}z^{4}+17a^{2}z^{4}+37z^{4}a^{-2}+9z^{4}a^{-4}+40z^{4}-4a^{3}z^{3}-18az^{3}-46z^{3}a^{-1}-46z^{3}a^{-3}-14z^{3}a^{-5}-6a^{4}z^{2}-34a^{2}z^{2}-73z^{2}a^{-2}-26z^{2}a^{-4}-75z^{2}+2a^{3}z+16az+37za^{-1}+39za^{-3}+16za^{-5}+4a^{4}+24a^{2}+60a^{-2}+23a^{-4}+58-2a^{3}z^{-1}-12az^{-1}-24a^{-1}z^{-1}-23a^{-3}z^{-1}-9a^{-5}z^{-1}-a^{4}z^{-2}-7a^{2}z^{-2}-19a^{-2}z^{-2}-7a^{-4}z^{-2}-18z^{-2}+a^{3}z^{-3}+5az^{-3}+9a^{-1}z^{-3}+7a^{-3}z^{-3}+2a^{-5}z^{-3}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                         0 8                   5 1   4 6                 1 4     3 4               4 1       3 2           4 1 1         4 0         1 7 4           2 -2       1   1             2 -4       1 3               -2 -6   1 1                   0 -8                         0 -10 1                       -1

Integral Khovanov Homology (db, data source)

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