L10n84

L10n83

L10n85

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Link Presentations

[edit Notes on L10n84's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(uv^{2}w^{2}+uv^{2}w+uv+vw^{3}+w^{2}+w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-12}-q^{-11}+2q^{-10}-q^{-9}+2q^{-8}-q^{-7}+q^{-6}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-2z^{2}a^{10}-2a^{10}z^{-2}-4a^{10}-z^{2}a^{8}+a^{8}z^{-2}+z^{6}a^{6}+6z^{4}a^{6}+9z^{2}a^{6}+3a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{6}-5a^{14}z^{4}+6a^{14}z^{2}-2a^{14}+a^{13}z^{7}-4a^{13}z^{5}+2a^{13}z^{3}+a^{13}z+a^{12}z^{8}-5a^{12}z^{6}+7a^{12}z^{4}-6a^{12}z^{2}-a^{12}z^{-2}+4a^{12}+2a^{11}z^{7}-11a^{11}z^{5}+17a^{11}z^{3}-10a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+17a^{10}z^{4}-20a^{10}z^{2}-2a^{10}z^{-2}+10a^{10}+a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}-a^{8}z^{4}+a^{8}z^{2}-a^{8}z^{-2}+2a^{8}-a^{7}z^{3}+a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+9a^{6}z^{2}-3a^{6}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                     1 1 -9               1 1     0 -11             1         1 -13           2 3 1       0 -15         1             1 -17       1 3 1           1 -19     1 1 1             1 -21     1                 1 -23 1 1                   0 -25 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $i=-3$ $r=-10$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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