L11a136

L11a135

L11a137

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Link Presentations

[edit Notes on L11a136's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3t(1)t(2)^{3}-4t(2)^{3}-8t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-8t(2)-4t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-4za^{5}-2a^{5}z^{-1}-z^{5}a^{3}+z^{3}a^{3}+3za^{3}+2a^{3}z^{-1}-z^{5}a-z^{3}a-2za-az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-12a^{8}z^{6}+13a^{8}z^{4}-5a^{8}z^{2}+4a^{7}z^{9}-15a^{7}z^{7}+16a^{7}z^{5}-8a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{10}-20a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+10a^{5}z^{9}-36a^{5}z^{7}+43a^{5}z^{5}-24a^{5}z^{3}+9a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+5a^{4}z^{8}-28a^{4}z^{6}+29a^{4}z^{4}-6a^{4}z^{2}-a^{4}+6a^{3}z^{9}-12a^{3}z^{7}+11a^{3}z^{5}-11a^{3}z^{3}+8a^{3}z-2a^{3}z^{-1}+8a^{2}z^{8}-13a^{2}z^{6}+7a^{2}z^{4}+z^{4}a^{-2}-2a^{2}z^{2}+8az^{7}-8az^{5}+4z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+3az-az^{-1}+7z^{6}-7z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   4 1   -3 0                 7 3     4 -2               8 5       -3 -4             8 6         2 -6           7 8           1 -8         6 8             -2 -10       4 8               4 -12     2 5                 -3 -14   1 4                   3 -16   2                     -2 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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