L11n113

L11n112

L11n114

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Link Presentations

[edit Notes on L11n113's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 6, -5, -9, 7, -2, 11, -8, 9, -7, 8, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(u-1)(v-1)^{3}}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}+{\frac {10}{q^{7/2}}}-{\frac {11}{q^{5/2}}}-2q^{3/2}+{\frac {10}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-11a^{3}z-6a^{3}z^{-1}+4az^{3}+8az+5az^{-1}-2za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-8a^{5}z^{7}-9a^{3}z^{7}-4az^{7}-a^{8}z^{6}+4a^{6}z^{6}+15a^{4}z^{6}+9a^{2}z^{6}-z^{6}+9a^{7}z^{5}+33a^{5}z^{5}+36a^{3}z^{5}+12az^{5}+3a^{8}z^{4}+8a^{6}z^{4}-3a^{4}z^{4}-10a^{2}z^{4}-2z^{4}-8a^{7}z^{3}-35a^{5}z^{3}-50a^{3}z^{3}-26az^{3}-3z^{3}a^{-1}-3a^{8}z^{2}-9a^{6}z^{2}-7a^{4}z^{2}+z^{2}+4a^{7}z+18a^{5}z+29a^{3}z+20az+5za^{-1}+a^{8}+3a^{6}+3a^{4}+a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-6a^{3}z^{-1}-5az^{-1}-2a^{-1}z^{-1}$ (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 4                   2 2 2                 2   -2 0               6 2   4 -2             6 4     -2 -4           5 4       1 -6         5 6         1 -8       4 5           -1 -10     2 5             3 -12   1 4               -3 -14   2                 2 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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