L11n141

L11n140

L11n142

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Link Presentations

[edit Notes on L11n141's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 X19,4,20,5
Gauss code	{1, -4, -3, 11, 5, -6}, {6, -1, -7, 3, 2, -5, -9, 10, -8, 7, 4, -2, -11, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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