L10n96

L10n95

L10n97

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Link Presentations

[edit Notes on L10n96's Link Presentations]

Planar diagram presentation	X6172 X2536 X13,15,14,20 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X11,19,12,18 X19,13,20,12 X17,9,18,14
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -8, 9, -3, 10}, {-7, 6, -10, 8, -9, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uvw^{2}x-uvwx^{2}+2uvwx-uvw-uvx+uv+uwx^{2}-uwx+ux+vw^{2}x-vwx+vw+w^{2}x^{2}-w^{2}x-wx^{2}+2wx-w-x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-q^{19/2}+2q^{17/2}-5q^{15/2}+5q^{13/2}-8q^{11/2}+6q^{9/2}-7q^{7/2}+3q^{5/2}-3q^{3/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}+3z^{3}a^{-3}-9z^{3}a^{-5}+3z^{3}a^{-7}+8za^{-3}-16za^{-5}+9za^{-7}-za^{-9}+5a^{-3}z^{-1}-12a^{-5}z^{-1}+9a^{-7}z^{-1}-2a^{-9}z^{-1}+a^{-3}z^{-3}-3a^{-5}z^{-3}+3a^{-7}z^{-3}-a^{-9}z^{-3}$ (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 -z^8 a^{-6} -z^8 a^{-8} -4 z^7 a^{-5} -6 z^7 a^{-7} -2 z^7 a^{-9} -3 z^6 a^{-4} -4 z^6 a^{-6} -3 z^6 a^{-8} -2 z^6 a^{-10} +15 z^5 a^{-5} +18 z^5 a^{-7} +2 z^5 a^{-9} -z^5 a^{-11} +9 z^4 a^{-4} +18 z^4 a^{-6} +13 z^4 a^{-8} +4 z^4 a^{-10} -6 z^3 a^{-3} -31 z^3 a^{-5} -26 z^3 a^{-7} +2 z^3 a^{-9} +3 z^3 a^{-11} -17 z^2 a^{-4} -33 z^2 a^{-6} -16 z^2 a^{-8} +13 z a^{-3} +28 z a^{-5} +21 z a^{-7} +3 z a^{-9} -3 z a^{-11} +13 a^{-4} +24 a^{-6} +11 a^{-8} - a^{-10} -6 a^{-3} z^{-1} -14 a^{-5} z^{-1} -12 a^{-7} z^{-1} -3 a^{-9} z^{-1} + a^{-11} z^{-1} -3 a^{-4} z^{-2} -6 a^{-6} z^{-2} -3 a^{-8} z^{-2} + a^{-3} z^{-3} +3 a^{-5} z^{-3} +3 a^{-7} z^{-3} + a^{-9} z^{-3} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 χ 20                 1 1 18               1   -1 16             4 1   3 14           4 4     0 12         4 1       3 10       3 5         2 8     4 3           1 6     4             4 4 3 3               0 2 3                 3

Integral Khovanov Homology (db, data source)

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

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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