L11n447

L11n446

L11n448

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Link Presentations

[edit Notes on L11n447's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,17,20,22 X21,9,22,16 X15,21,16,20 X17,12,18,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {-9, 5, -6, 8, -7, 6}, {11, -2, -5, 9, 4, -3, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)(x-1)\left(-uv+ux^{2}-ux+u+vx^{2}-vx+v-x^{2}\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+2{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}+a^{9}z^{-3}-4za^{7}-7a^{7}z^{-1}-3a^{7}z^{-3}+5z^{3}a^{5}+13za^{5}+12a^{5}z^{-1}+3a^{5}z^{-3}-2z^{5}a^{3}-7z^{3}a^{3}-11za^{3}-7a^{3}z^{-1}-a^{3}z^{-3}+2z^{3}a+2za+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+10a^{9}z^{3}-a^{9}z^{-3}-10a^{9}z+5a^{9}z^{-1}+2a^{8}z^{8}-7a^{8}z^{6}+5a^{8}z^{4}+6a^{8}z^{2}+3a^{8}z^{-2}-9a^{8}+a^{7}z^{9}+5a^{7}z^{7}-34a^{7}z^{5}+55a^{7}z^{3}-3a^{7}z^{-3}-38a^{7}z+14a^{7}z^{-1}+8a^{6}z^{8}-22a^{6}z^{6}+a^{6}z^{4}+27a^{6}z^{2}+6a^{6}z^{-2}-21a^{6}+a^{5}z^{9}+15a^{5}z^{7}-65a^{5}z^{5}+83a^{5}z^{3}-3a^{5}z^{-3}-54a^{5}z+18a^{5}z^{-1}+6a^{4}z^{8}-8a^{4}z^{6}-18a^{4}z^{4}+33a^{4}z^{2}+3a^{4}z^{-2}-18a^{4}+11a^{3}z^{7}-35a^{3}z^{5}+45a^{3}z^{3}-a^{3}z^{-3}-31a^{3}z+11a^{3}z^{-1}+7a^{2}z^{6}-14a^{2}z^{4}+15a^{2}z^{2}-6a^{2}+az^{5}+7az^{3}-5az+2az^{-1}+3z^{2}-1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   2 -2 0                 4   4 -2               6 4   -2 -4             5 3 1   3 -6           4 6       2 -8         8 5         3 -10       5 10           5 -12     1 2             -1 -14   1 5               4 -16   1                 -1 -18 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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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