L11a482

L11a481

L11a483

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Link Presentations

[edit Notes on L11a482's Link Presentations]

Planar diagram presentation	X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X20,14,21,13 X18,7,19,8 X8,17,9,18 X22,20,17,19 X14,22,15,21
Gauss code	{1, -2, 3, -6}, {9, -8, 10, -7, 11, -10}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-2)(t(3)-1)(2t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{2}+5q-11+17q^{-1}-20q^{-2}+25q^{-3}-22q^{-4}+19q^{-5}-13q^{-6}+7q^{-7}-3q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{6}a^{4}+z^{4}a^{4}+z^{2}a^{4}-2a^{4}z^{-2}-a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+2a^{2}-z^{4}$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+4z^{8}a^{8}-5z^{6}a^{8}-4z^{4}a^{8}+6z^{2}a^{8}-a^{8}+4z^{9}a^{7}-z^{7}a^{7}-11z^{5}a^{7}+13z^{3}a^{7}-6za^{7}+2z^{10}a^{6}+8z^{8}a^{6}-20z^{6}a^{6}+11z^{4}a^{6}+z^{2}a^{6}+a^{6}z^{-2}-a^{6}+12z^{9}a^{5}-17z^{7}a^{5}+z^{5}a^{5}+8z^{3}a^{5}-2za^{5}-2a^{5}z^{-1}+2z^{10}a^{4}+17z^{8}a^{4}-39z^{6}a^{4}+23z^{4}a^{4}-z^{2}a^{4}+2a^{4}z^{-2}-2a^{4}+8z^{9}a^{3}-2z^{7}a^{3}-12z^{5}a^{3}+4z^{3}a^{3}+2za^{3}-2a^{3}z^{-1}+13z^{8}a^{2}-20z^{6}a^{2}+7z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}-2a^{2}+11z^{7}a-15z^{5}a+2z^{3}a+5z^{6}-4z^{4}+z^{5}a^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   7 1   -6 -1                 10 4     6 -3               12 9       -3 -5             13 8         5 -7           9 12           3 -9         10 13             -3 -11       5 11               6 -13     2 8                 -6 -15   1 5                   4 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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