L11a115

L11a114

L11a116

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Link Presentations

[edit Notes on L11a115's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-7uv^{2}+10uv-4u-4v^{3}+10v^{2}-7v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {14}{q^{7/2}}}+q^{5/2}-{\frac {15}{q^{5/2}}}-4q^{3/2}+{\frac {13}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-3a^{7}z-2a^{7}z^{-1}+3a^{5}z^{3}+3a^{5}z+a^{5}z^{-1}-a^{3}z^{5}+a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}-2az-az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+8a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+9a^{8}z^{4}-4a^{8}z^{2}+a^{8}+2a^{7}z^{9}-4a^{7}z^{7}-7a^{7}z^{5}+18a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+3a^{6}z^{8}-20a^{6}z^{6}+25a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+6a^{5}z^{9}-15a^{5}z^{7}+4a^{5}z^{5}+11a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+a^{4}z^{10}+8a^{4}z^{8}-29a^{4}z^{6}+30a^{4}z^{4}-12a^{4}z^{2}+2a^{4}+4a^{3}z^{9}-2a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}+7a^{2}z^{8}-10a^{2}z^{6}+6a^{2}z^{4}+z^{4}a^{-2}-4a^{2}z^{2}+a^{2}+8az^{7}-9az^{5}+4z^{5}a^{-1}-3z^{3}a^{-1}+3az-az^{-1}+7z^{6}-7z^{4}$ (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               7 5       -2 -4             8 6         2 -6           6 7           1 -8         5 8             -3 -10       4 7               3 -12     1 4                 -3 -14   1 4                   3 -16   1                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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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