L11n132

L11n131

L11n133

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Link Presentations

[edit Notes on L11n132's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, -4, -2, 10, 8, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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