L11n272

L11n271

L11n273

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Link Presentations

[edit Notes on L11n272's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(3)^{3}-2t(1)t(3)^{2}-2t(2)t(3)^{2}-t(3)^{2}+2t(1)t(3)+t(1)t(2)t(3)+2t(2)t(3)-2t(1)t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-q^{2}+2q+1+q^{-1}+q^{-2}-q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}z^{-2}+6a^{4}-a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}-3a^{2}+2a^{-2}-z^{4}-4z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+6a^{6}z^{4}-2a^{6}z^{2}-a^{6}z^{-2}+a^{6}+3a^{5}z^{7}-18a^{5}z^{5}+33a^{5}z^{3}-27a^{5}z+8a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+5a^{4}z^{4}-4a^{4}z^{2}-3a^{4}z^{-2}+5a^{4}+3a^{3}z^{7}-21a^{3}z^{5}+42a^{3}z^{3}-34a^{3}z+10a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+z^{6}a^{-2}+7a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+6z^{2}a^{-2}-2a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}-4a^{-2}+2az^{7}+z^{7}a^{-1}-13az^{5}-4z^{5}a^{-1}+20az^{3}-10az+4za^{-1}+2az^{-1}-2a^{-1}z^{-1}+z^{8}-5z^{6}+3z^{4}+5z^{2}+z^{-2}-3$ (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 1 2 3 4 χ 7                       1 1 5                     1 1 0 3                   1     1 1               3 1 1     3 -1             2 5 1       2 -3           1   1         2 -5         1 3 2           0 -7       2 1               1 -9       1 1               0 -11   1 2                   -1 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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