L10a98

L10a97

L10a99

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Link Presentations

[edit Notes on L10a98's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,7,19,8 X20,15,9,16 X16,19,17,20 X8,9,1,10 X4,13,5,14 X6,17,7,18
Gauss code	{1, -2, 3, -9, 4, -10, 5, -8}, {8, -1, 2, -3, 9, -4, 6, -7, 10, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{27/2}}}+{\frac {2}{q^{25/2}}}-{\frac {3}{q^{23/2}}}+{\frac {4}{q^{21/2}}}-{\frac {5}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}z^{5}+4a^{11}z^{3}+3a^{11}z-a^{9}z^{7}-5a^{9}z^{5}-6a^{9}z^{3}+a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-11a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+2z^{2}a^{16}-2z^{5}a^{15}+z^{3}a^{15}-2z^{6}a^{14}+2z^{4}a^{14}-z^{2}a^{14}-2z^{7}a^{13}+4z^{5}a^{13}-3z^{3}a^{13}+za^{13}-2z^{8}a^{12}+7z^{6}a^{12}-9z^{4}a^{12}+5z^{2}a^{12}-z^{9}a^{11}+3z^{7}a^{11}-3z^{5}a^{11}+5z^{3}a^{11}-3za^{11}-3z^{8}a^{10}+13z^{6}a^{10}-15z^{4}a^{10}+5z^{2}a^{10}-z^{9}a^{9}+4z^{7}a^{9}-3z^{5}a^{9}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-2z^{4}a^{8}-3z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+7za^{7}-a^{7}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 2     2 -12               1 1     0 -14             3 2       1 -16           2 1         -1 -18         3 3           0 -20       1 2             1 -22     2 3               -1 -24   1 2                 1 -26   1                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

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

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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