L11a386

L11a385

L11a387

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Link Presentations

[edit Notes on L11a386's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-3t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+3t(3)^{3}+7t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+7t(2)t(3)^{2}-5t(3)^{2}-7t(1)t(3)+5t(1)t(2)t(3)-7t(2)t(3)+4t(3)+3t(1)-3t(1)t(2)+3t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-2q^{-5}+4q^{4}+7q^{-4}-10q^{3}-11q^{-3}+15q^{2}+19q^{-2}-20q-20q^{-1}+22$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-a^{4}z^{-2}-z^{2}a^{-4}-4a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+3a^{2}z^{2}-2a^{2}z^{-2}-a^{-2}z^{-2}-a^{2}-3a^{-2}-z^{6}+4z^{2}+3z^{-2}+7$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+8az^{9}+5z^{9}a^{-1}+3a^{4}z^{8}+12a^{2}z^{8}+10z^{8}a^{-2}+19z^{8}+2a^{5}z^{7}+a^{3}z^{7}+2az^{7}+12z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-3a^{4}z^{6}-30a^{2}z^{6}-13z^{6}a^{-2}+4z^{6}a^{-4}-43z^{6}-4a^{5}z^{5}-12a^{3}z^{5}-39az^{5}-47z^{5}a^{-1}-15z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-4a^{4}z^{4}+26a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}a^{-4}+34z^{4}+16a^{3}z^{3}+53az^{3}+50z^{3}a^{-1}+12z^{3}a^{-3}-z^{3}a^{-5}+6a^{6}z^{2}+6a^{4}z^{2}-11a^{2}z^{2}-2z^{2}a^{-2}+z^{2}a^{-4}-14z^{2}+4a^{5}z-10a^{3}z-34az-27za^{-1}-7za^{-3}-4a^{6}-3a^{4}+4a^{2}+a^{-2}+5-2a^{5}z^{-1}+2a^{3}z^{-1}+10az^{-1}+8a^{-1}z^{-1}+2a^{-3}z^{-1}+a^{6}z^{-2}+a^{4}z^{-2}-2a^{2}z^{-2}-a^{-2}z^{-2}-3z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 8 3     5 3               12 7       -5 1             10 8         2 -1           11 13           2 -3         8 9             -1 -5       3 11               8 -7     4 8                 -4 -9   1 6                   5 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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