L11n74

L11n73

L11n75

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Link Presentations

[edit Notes on L11n74's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X13,21,14,20 X15,5,16,22 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{5}+uv^{4}-uv^{3}+uv^{2}-u-v^{7}+v^{5}-v^{4}+v^{3}-v^{2}}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$3q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{q^{3/2}}}+q^{17/2}-q^{15/2}+2q^{13/2}-3q^{11/2}-{\sqrt {q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+2z^{3}a^{-5}+5za^{-5}+3a^{-5}z^{-1}-z^{7}a^{-3}-8z^{5}a^{-3}-20z^{3}a^{-3}-19za^{-3}-7a^{-3}z^{-1}+z^{5}a^{-1}+6z^{3}a^{-1}+10za^{-1}+4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-3z^{2}a^{-10}+a^{-10}+z^{5}a^{-9}-2z^{3}a^{-9}+z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+z^{7}a^{-7}-4z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+z^{6}a^{-6}-2z^{4}a^{-6}+3z^{2}a^{-6}+z^{5}a^{-5}-3z^{3}a^{-5}+7za^{-5}-3a^{-5}z^{-1}+z^{8}a^{-4}-9z^{6}a^{-4}+26z^{4}a^{-4}-26z^{2}a^{-4}+7a^{-4}+z^{9}a^{-3}-10z^{7}a^{-3}+34z^{5}a^{-3}-49z^{3}a^{-3}+30za^{-3}-7a^{-3}z^{-1}+z^{8}a^{-2}-9z^{6}a^{-2}+25z^{4}a^{-2}-25z^{2}a^{-2}+7a^{-2}+z^{9}a^{-1}-9z^{7}a^{-1}+28z^{5}a^{-1}-37z^{3}a^{-1}+21za^{-1}-4a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                         0 14                   2 1   -1 12                 1       1 10               2 2       0 8           1 1 1         -1 6           1 2           1 4       1 2 1             0 2         2               2 0     1                   1 -2 1                       1 -4 1                       1

Integral Khovanov Homology (db, data source)

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

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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