L11n379

L11n378

L11n380

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Link Presentations

[edit Notes on L11n379's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X22,18,19,17 X20,12,21,11 X12,20,13,19 X18,22,5,21 X9,16,10,17 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)(t(3)+1)\left(t(1)t(3)^{3}+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{2}+1+q^{-1}+q^{-2}+q^{-3}-q^{-4}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-a^{6}+a^{4}z^{4}+6a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{6}-7a^{2}z^{4}-15a^{2}z^{2}-5a^{2}z^{-2}-13a^{2}+z^{4}+5z^{2}+2z^{-2}+6$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+a^{3}z^{7}+az^{7}-a^{4}z^{6}-9a^{2}z^{6}-8z^{6}-a^{5}z^{5}-9a^{3}z^{5}-8az^{5}+7a^{4}z^{4}+28a^{2}z^{4}+21z^{4}+6a^{5}z^{3}+26a^{3}z^{3}+20az^{3}-15a^{4}z^{2}-37a^{2}z^{2}-22z^{2}-9a^{5}z-27a^{3}z-18az+a^{6}+12a^{4}+21a^{2}+11+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 5                 1 1 3                 1 1 1             1     1 -1         2         2 -3         3 1       2 -5     2             2 -7     1 1           0 -9 1 2               -1 -11 1 1               0

Integral Khovanov Homology (db, data source)

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

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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