L10a150

L10a149

L10a151

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Link Presentations

[edit Notes on L10a150's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)\left(2vw^{2}-2vw+v-w^{2}+2w-2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+8q^{8}-11q^{7}+13q^{6}-13q^{5}+13q^{4}-8q^{3}+6q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+z^{4}a^{-8}+3z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-8}+3a^{-2}-4a^{-4}+a^{-6}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}+4z^{5}a^{-11}-2z^{3}a^{-11}+8z^{6}a^{-10}-9z^{4}a^{-10}+3z^{2}a^{-10}-a^{-10}+9z^{7}a^{-9}-12z^{5}a^{-9}+5z^{3}a^{-9}-za^{-9}+5z^{8}a^{-8}+2z^{6}a^{-8}-15z^{4}a^{-8}+11z^{2}a^{-8}-3a^{-8}+z^{9}a^{-7}+13z^{7}a^{-7}-29z^{5}a^{-7}+16z^{3}a^{-7}-3za^{-7}+7z^{8}a^{-6}-7z^{6}a^{-6}-13z^{4}a^{-6}+18z^{2}a^{-6}+a^{-6}z^{-2}-7a^{-6}+z^{9}a^{-5}+6z^{7}a^{-5}-18z^{5}a^{-5}+11z^{3}a^{-5}+za^{-5}-2a^{-5}z^{-1}+2z^{8}a^{-4}-12z^{4}a^{-4}+16z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+2z^{7}a^{-3}-5z^{5}a^{-3}+2z^{3}a^{-3}+3za^{-3}-2a^{-3}z^{-1}+z^{6}a^{-2}-4z^{4}a^{-2}+6z^{2}a^{-2}+a^{-2}z^{-2}-4a^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   3   -3 17                 5 1   4 15               6 3     -3 13             7 5       2 11           8 8         0 9         5 5           0 7       3 8             5 5     3 5               -2 3   1 5                 4 1   1                   -1 -1 1                     1

Integral Khovanov Homology (db, data source)

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

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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