L11n306

L11n305

L11n307

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Link Presentations

[edit Notes on L11n306's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X15,22,16,11 X13,20,14,21 X21,14,22,15 X17,8,18,9 X7,16,8,17 X9,18,10,19 X19,10,20,5 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 9}, {-11, 2, -4, 5, -3, 7, -6, 8, -9, 4, -5, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{4}-uv^{2}w^{3}+uv^{2}w^{2}-uv^{2}w+uvw-uv+u-v^{2}w^{4}+vw^{4}-vw^{3}+w^{3}-w^{2}+w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-3q^{-8}+4q^{-7}-4q^{-6}+5q^{-5}-3q^{-4}+4q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}z^{-2}-2a^{10}+a^{8}z^{6}+6a^{8}z^{4}+12a^{8}z^{2}+4a^{8}z^{-2}+11a^{8}-a^{6}z^{8}-7a^{6}z^{6}-18a^{6}z^{4}-24a^{6}z^{2}-5a^{6}z^{-2}-18a^{6}+a^{4}z^{6}+6a^{4}z^{4}+12a^{4}z^{2}+2a^{4}z^{-2}+9a^{4}$ (db)
Kauffman polynomial	$za^{13}+2z^{2}a^{12}-a^{12}+z^{5}a^{11}-z^{3}a^{11}-2za^{11}+a^{11}z^{-1}+3z^{6}a^{10}-10z^{4}a^{10}+6z^{2}a^{10}-a^{10}z^{-2}+5z^{7}a^{9}-23z^{5}a^{9}+31z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+4z^{8}a^{8}-19z^{6}a^{8}+27z^{4}a^{8}-21z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+z^{9}a^{7}+z^{7}a^{7}-25z^{5}a^{7}+50z^{3}a^{7}-35za^{7}+9a^{7}z^{-1}+5z^{8}a^{6}-29z^{6}a^{6}+55z^{4}a^{6}-46z^{2}a^{6}-5a^{6}z^{-2}+22a^{6}+z^{9}a^{5}-4z^{7}a^{5}-z^{5}a^{5}+18z^{3}a^{5}-19za^{5}+5a^{5}z^{-1}+z^{8}a^{4}-7z^{6}a^{4}+18z^{4}a^{4}-21z^{2}a^{4}-2a^{4}z^{-2}+11a^{4}$ (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 χ -1                   1 1 -3                     0 -5               4 1   3 -7             1 2     1 -9           4 2       2 -11       1 2 2         1 -13       3 3           0 -15   1 2 2             1 -17   2 3               -1 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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