L10a137

L10a136

L10a138

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Link Presentations

[edit Notes on L10a137's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X4,17,1,18 X12,20,5,19 X20,12,13,11 X10,14,11,13
Gauss code	{1, -4, 3, -7}, {2, -1, 5, -3, 6, -10, 9, -8}, {10, -2, 4, -5, 7, -6, 8, -9}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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