L10a23

L10a22

L10a24

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Link Presentations

[edit Notes on L10a23's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,12,15,11 X10,4,11,3 X12,5,13,6 X20,13,5,14 X2,16,3,15
Gauss code	{1, -10, 7, -5}, {8, -1, 2, -4, 3, -7, 6, -8, 9, -6, 10, -2, 4, -3, 5, -9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z-a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-2a^{3}z+az^{5}+2az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-2a^{9}z^{3}+6a^{8}z^{6}-8a^{8}z^{4}+6a^{8}z^{2}-2a^{8}+7a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-2a^{6}z^{6}-9a^{6}z^{4}+12a^{6}z^{2}-5a^{6}+2a^{5}z^{9}+7a^{5}z^{7}-23a^{5}z^{5}+20a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+10a^{4}z^{8}-23a^{4}z^{6}+12a^{4}z^{4}+3a^{4}z^{2}-3a^{4}+2a^{3}z^{9}+4a^{3}z^{7}-22a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+5a^{2}z^{8}-14a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+a^{2}+4az^{7}-12az^{5}+8az^{3}+az-az^{-1}+z^{6}-2z^{4}$ (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 χ 4                     1 -1 2                   3   3 0                 3 1   -2 -2               7 3     4 -4             7 5       -2 -6           7 5         2 -8         5 7           2 -10       5 7             -2 -12     2 5               3 -14   1 5                 -4 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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