L11n193

L11n192

L11n194

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Link Presentations

[edit Notes on L11n193's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X19,11,20,10
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, -2, 11, 9, -10, 7, -8, 5, -4, 6, -7, -11, 2, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+t(1)^{2}t(2)^{3}-5t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-5t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {10}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}+z^{5}a^{7}+z^{3}a^{7}+za^{7}+a^{7}z^{-1}+z^{5}a^{5}-2za^{5}-a^{5}z^{-1}-2z^{3}a^{3}-2za^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+4a^{11}z^{7}-11a^{11}z^{5}+7a^{11}z^{3}+5a^{10}z^{8}-12a^{10}z^{6}+4a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{9}+4a^{9}z^{7}-20a^{9}z^{5}+12a^{9}z^{3}-2a^{9}z+9a^{8}z^{8}-19a^{8}z^{6}+8a^{8}z^{4}-a^{8}z^{2}+2a^{7}z^{9}+3a^{7}z^{7}-9a^{7}z^{5}+a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+4a^{6}z^{8}-5a^{6}z^{6}+6a^{6}z^{4}-4a^{6}z^{2}+a^{6}+3a^{5}z^{7}-a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+a^{4}z^{6}+4a^{4}z^{4}-2a^{4}z^{2}+3a^{3}z^{3}-2a^{3}z$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 4 1 -3 -6               5 1   4 -8             6 4     -2 -10           7 5       2 -12         5 6         1 -14       5 7           -2 -16     3 6             3 -18   1 4               -3 -20   3                 3 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

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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