L11n451

L11n450

L11n452

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Link Presentations

[edit Notes on L11n451's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X15,8,16,5 X7,14,8,15 X16,19,17,20 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {9, -3, 5, -4}, {-11, 2, 3, -6, 4, 8, -7, -9, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)(x-1)^{2}(ux-v)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+a^{5}z^{-3}+7a^{5}z+5a^{5}z^{-1}-2a^{3}z^{5}-9a^{3}z^{3}-3a^{3}z^{-3}-15a^{3}z-10a^{3}z^{-1}+az^{5}+7az^{3}+3az^{-3}-z^{3}a^{-1}-a^{-1}z^{-3}+12az+9az^{-1}-3za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}+a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-9z^{3}a^{7}+5za^{7}-2a^{7}z^{-1}-2z^{8}a^{6}+z^{6}a^{6}+15z^{4}a^{6}-15z^{2}a^{6}+6a^{6}-9z^{7}a^{5}+35z^{5}a^{5}-43z^{3}a^{5}+31za^{5}-11a^{5}z^{-1}+a^{5}z^{-3}-2z^{8}a^{4}+z^{6}a^{4}+19z^{4}a^{4}-33z^{2}a^{4}-3a^{4}z^{-2}+18a^{4}-7z^{7}a^{3}+39z^{5}a^{3}-73z^{3}a^{3}+54za^{3}-18a^{3}z^{-1}+3a^{3}z^{-3}-z^{8}a^{2}+5z^{6}a^{2}+2z^{4}a^{2}-27z^{2}a^{2}-6a^{2}z^{-2}+21a^{2}-2z^{7}a+20z^{5}a-49z^{3}a+38za-14az^{-1}+3az^{-3}-z^{8}+6z^{6}-5z^{4}-6z^{2}-3z^{-2}+9-z^{7}a^{-1}+6z^{5}a^{-1}-10z^{3}a^{-1}+10za^{-1}-5a^{-1}z^{-1}+a^{-1}z^{-3}$ (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 3 4 χ 6                       1 1 4                         0 2                   1 1   0 0               6 1       5 -2             4 6 1       1 -4           4 2 2         4 -6         2 4             2 -8       5 4 1             2 -10     2 4                 2 -12   1 3                   -2 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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