L11a414

L11a413

L11a415

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Link Presentations

[edit Notes on L11a414's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{2}t(3)^{4}-2t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}+t(2)t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-11q^{5}+12q^{4}-11q^{3}+10q^{2}-6q+5-q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-11z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-21z^{2}a^{-2}+18z^{2}a^{-4}-5z^{2}a^{-6}+5z^{2}-17a^{-2}+13a^{-4}-3a^{-6}+7-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+5z^{9}a^{-3}+4z^{9}a^{-5}-z^{8}a^{-2}+6z^{8}a^{-4}+8z^{8}a^{-6}+z^{8}-3z^{7}a^{-1}-16z^{7}a^{-3}-3z^{7}a^{-5}+10z^{7}a^{-7}-13z^{6}a^{-2}-32z^{6}a^{-4}-17z^{6}a^{-6}+9z^{6}a^{-8}-7z^{6}-4z^{5}a^{-1}-z^{5}a^{-3}-23z^{5}a^{-5}-20z^{5}a^{-7}+6z^{5}a^{-9}+36z^{4}a^{-2}+37z^{4}a^{-4}+3z^{4}a^{-6}-13z^{4}a^{-8}+3z^{4}a^{-10}+18z^{4}+20z^{3}a^{-1}+39z^{3}a^{-3}+33z^{3}a^{-5}+9z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-39z^{2}a^{-2}-20z^{2}a^{-4}+3z^{2}a^{-6}+5z^{2}a^{-8}-21z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 5 2     3 11               6 4       -2 9             6 5         1 7           7 8           1 5         3 4             -1 3       4 8               4 1     1 2                 -1 -1     4                   4 -3 1 1                     0 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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