L11n311

L11n310

L11n312

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Link Presentations

[edit Notes on L11n311's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,20,10,21 X15,5,16,10 X19,15,20,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -5, 6, -7, 8}, {11, -2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-uv^{2}w+uvw^{3}-2uvw^{2}+uvw+uv+uw-u+v^{2}w^{4}-v^{2}w^{3}-vw^{4}-vw^{3}+2vw^{2}-vw+w^{3}-w^{2}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+q^{-5}+2q^{4}-q^{-4}-3q^{3}+2q^{-3}+3q^{2}+q^{-2}-3q+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{6}-2a^{2}z^{4}+7z^{4}+a^{4}z^{2}-10a^{2}z^{2}-3z^{2}a^{-2}-z^{2}a^{-4}+14z^{2}+3a^{4}-12a^{2}-4a^{-2}+13+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{4}z^{8}-7a^{4}z^{6}+2z^{6}a^{-4}+16a^{4}z^{4}-6z^{4}a^{-4}-17a^{4}z^{2}+2z^{2}a^{-4}-2a^{4}z^{-2}+9a^{4}+a^{3}z^{9}-6a^{3}z^{7}+2z^{7}a^{-3}+7a^{3}z^{5}-7z^{5}a^{-3}+6a^{3}z^{3}+6z^{3}a^{-3}-13a^{3}z-4za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+z^{8}a^{-2}-23a^{2}z^{6}-4z^{6}a^{-2}+53a^{2}z^{4}+6z^{4}a^{-2}-49a^{2}z^{2}-8z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+21a^{2}+5a^{-2}+az^{9}-5az^{7}+3z^{7}a^{-1}-3az^{5}-18z^{5}a^{-1}+30az^{3}+33z^{3}a^{-1}-29az-21za^{-1}+9az^{-1}+5a^{-1}z^{-1}+3z^{8}-22z^{6}+49z^{4}-42z^{2}-4z^{-2}+18$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     1   1 7                   2 1   -1 5               1 2 1     0 3             1 3 2       0 1           1 3 2         0 -1         1 2 4           3 -3       1 3 1             1 -5     1 1 3               3 -7   1 2                   1 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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