L10n34

L10n33

L10n35

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Link Presentations

[edit Notes on L10n34's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,8,17,7 X17,20,18,5 X11,19,12,18 X19,11,20,10 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 7, 6, -5, -2, 10, -7, 8, -3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+3q^{11/2}-4q^{9/2}+4q^{7/2}-4q^{5/2}+3q^{3/2}-3{\sqrt {q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{3}a^{-5}+a^{-5}z^{-1}-2z^{3}a^{-3}-4za^{-3}-3a^{-3}z^{-1}+3za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-3z^{7}a^{-5}-2z^{7}a^{-7}+3z^{6}a^{-4}+2z^{6}a^{-6}-z^{6}a^{-8}+z^{5}a^{-3}+9z^{5}a^{-5}+8z^{5}a^{-7}-3z^{4}a^{-2}-6z^{4}a^{-4}+z^{4}a^{-6}+4z^{4}a^{-8}+3z^{3}a^{-3}-5z^{3}a^{-5}-8z^{3}a^{-7}+4z^{2}a^{-2}+9z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}a^{-8}-5za^{-1}-7za^{-3}+2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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 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   \ 0 1 2 3 4 5 6 7 χ 16               1 -1 14             1   1 12           2 1   -1 10         2 1     1 8       2 2       0 6     2 2         0 4   1 2           1 2 2 2             0 0 3               3

Integral Khovanov Homology (db, data source)

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

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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