L11a78

L11a77

L11a79

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Link Presentations

[edit Notes on L11a78's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X22,15,5,16 X16,7,17,8 X8,21,9,22 X18,10,19,9 X20,18,21,17 X10,20,11,19 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 7, -9, 3, -2, 11, -3, 4, -5, 8, -7, 9, -8, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+15q^{7/2}-20q^{5/2}+22q^{3/2}-22{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-9z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+8az-9za^{-1}+3za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-5z^{4}a^{-6}+10z^{7}a^{-5}-15z^{5}a^{-5}+5z^{3}a^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-10z^{6}a^{-4}-3a^{4}z^{4}-4z^{4}a^{-4}+3a^{4}z^{2}+5z^{2}a^{-4}-a^{4}-a^{-4}+5z^{9}a^{-3}+3a^{3}z^{7}+11z^{7}a^{-3}-8a^{3}z^{5}-35z^{5}a^{-3}+8a^{3}z^{3}+23z^{3}a^{-3}-4a^{3}z-6za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+z^{10}a^{-2}+4a^{2}z^{8}+18z^{8}a^{-2}-5a^{2}z^{6}-32z^{6}a^{-2}-5a^{2}z^{4}+6z^{4}a^{-2}+10a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+3az^{9}+8z^{9}a^{-1}+5az^{7}+3z^{7}a^{-1}-25az^{5}-36z^{5}a^{-1}+28az^{3}+38z^{3}a^{-1}-15az-17za^{-1}+4az^{-1}+4a^{-1}z^{-1}+z^{10}+12z^{8}-23z^{6}+3z^{4}+14z^{2}-7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   6 1   5 8                 9 4     -5 6               11 6       5 4             11 9         -2 2           11 11           0 0         8 13             5 -2       5 9               -4 -4     2 8                 6 -6   1 5                   -4 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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