L11n440

L11n439

L11n441

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Link Presentations

[edit Notes on L11n440's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X19,13,20,22 X15,21,16,20 X21,17,22,16 X12,17,9,18
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 $u$, $v$, $w$, ...)	${\frac {uvwx^{2}-uvwx-2uvx^{2}+2uvx-uv+uw+ux^{2}-ux-vwx^{2}+vwx+vx^{3}-wx^{3}+2wx^{2}-2wx-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+5q^{11/2}-6q^{9/2}+5q^{7/2}-7q^{5/2}+3q^{3/2}-5{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+2z^{5}a^{-3}+az^{3}-8z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+4az-17za^{-1}+21za^{-3}-9za^{-5}+za^{-7}+5az^{-1}-17a^{-1}z^{-1}+21a^{-3}z^{-1}-11a^{-5}z^{-1}+2a^{-7}z^{-1}+2az^{-3}-7a^{-1}z^{-3}+9a^{-3}z^{-3}-5a^{-5}z^{-3}+a^{-7}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+6z^{2}a^{-8}+a^{-8}z^{-2}-4a^{-8}+2z^{7}a^{-7}-6z^{5}a^{-7}+4z^{3}a^{-7}-a^{-7}z^{-3}-2za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}+4z^{6}a^{-6}-26z^{4}a^{-6}+34z^{2}a^{-6}+7a^{-6}z^{-2}-24a^{-6}+7z^{7}a^{-5}-24z^{5}a^{-5}+23z^{3}a^{-5}-5a^{-5}z^{-3}-16za^{-5}+12a^{-5}z^{-1}+z^{8}a^{-4}+7z^{6}a^{-4}-43z^{4}a^{-4}+75z^{2}a^{-4}+18a^{-4}z^{-2}-58a^{-4}+5z^{7}a^{-3}-23z^{5}a^{-3}+41z^{3}a^{-3}-9a^{-3}z^{-3}-37za^{-3}+24a^{-3}z^{-1}+5z^{6}a^{-2}-32z^{4}a^{-2}+73z^{2}a^{-2}+19a^{-2}z^{-2}-60a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-12z^{5}a^{-1}+15az^{3}+37z^{3}a^{-1}-2az^{-3}-7a^{-1}z^{-3}-16az-39za^{-1}+9az^{-1}+23a^{-1}z^{-1}+z^{6}-11z^{4}+26z^{2}+7z^{-2}-23$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                   4 1   -3 10                 2 1     1 8               4 5       1 6           1 4 1         2 4             4           4 2       1 7 4             2 0         6               6 -2     1                   1 -4 1                       1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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