L11n116

L11n115

L11n117

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Link Presentations

[edit Notes on L11n116's Link Presentations]

Planar diagram presentation	X6172 X7,17,8,16 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X15,9,16,8
Gauss code	{1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, 5, -3, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-7t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-7t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {14}{q^{13/2}}}-{\frac {11}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-1}+a^{7}z^{5}-3a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+5a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-3a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+4a^{11}z^{7}-11a^{11}z^{5}+8a^{11}z^{3}+5a^{10}z^{8}-11a^{10}z^{6}+4a^{10}z^{4}+a^{10}z^{2}-a^{10}+2a^{9}z^{9}+7a^{9}z^{7}-28a^{9}z^{5}+19a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+11a^{8}z^{8}-22a^{8}z^{6}+7a^{8}z^{4}+2a^{8}z^{2}-3a^{8}+2a^{7}z^{9}+10a^{7}z^{7}-31a^{7}z^{5}+27a^{7}z^{3}-12a^{7}z+3a^{7}z^{-1}+6a^{6}z^{8}-7a^{6}z^{6}+a^{6}z^{4}+5a^{6}z^{2}-3a^{6}+7a^{5}z^{7}-14a^{5}z^{5}+22a^{5}z^{3}-15a^{5}z+4a^{5}z^{-1}+3a^{4}z^{6}+3a^{4}z^{2}-2a^{4}+6a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   3 3 -4                 4 1 -3 -6               7 2   5 -8             6 4     -2 -10           8 7       1 -12         7 7         0 -14       4 7           -3 -16     3 7             4 -18   1 4               -3 -20   3                 3 -22 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$

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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