L10n68

L10n67

L10n69

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Link Presentations

[edit Notes on L10n68's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)+t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+3q^{-5}-2q^{-6}+4q^{-7}-3q^{-8}+3q^{-9}-2q^{-10}+q^{-11}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}-a^{10}z^{4}-4a^{10}z^{2}+a^{10}z^{-2}-2a^{10}+a^{8}z^{6}+4a^{8}z^{4}+2a^{8}z^{2}-2a^{8}z^{-2}-3a^{8}+a^{6}z^{6}+5a^{6}z^{4}+7a^{6}z^{2}+a^{6}z^{-2}+4a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{2}-a^{14}+2a^{13}z^{3}-2a^{13}z+a^{12}z^{6}-3a^{12}z^{4}+5a^{12}z^{2}-a^{12}+2a^{11}z^{7}-9a^{11}z^{5}+14a^{11}z^{3}-6a^{11}z+a^{10}z^{8}-3a^{10}z^{6}+a^{10}z^{4}+3a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+3a^{9}z^{7}-12a^{9}z^{5}+11a^{9}z^{3}-2a^{9}z^{-1}+a^{8}z^{8}-3a^{8}z^{6}-a^{8}z^{4}+6a^{8}z^{2}+2a^{8}z^{-2}-6a^{8}+a^{7}z^{7}-3a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-5a^{6}z^{4}+7a^{6}z^{2}+a^{6}z^{-2}-4a^{6}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                 1 1 -7               1 1 0 -9             2     2 -11         1 1 1     1 -13         4 2       2 -15     1 1 1         1 -17     3 3           0 -19   1 2             1 -21   1               -1 -23 1                 1

Integral Khovanov Homology (db, data source)

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

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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