L11a351

L11a350

L11a352

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Link Presentations

[edit Notes on L11a351's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,9,19,10 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X8394 X16,21,17,22 X4,18,5,17 X10,13,1,14 X2,19,3,20
Gauss code	{1, -11, 7, -9, 3, -4, 6, -7, 2, -10}, {4, -1, 10, -3, 5, -8, 9, -2, 11, -6, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+2t(1)^{2}-2t(2)^{2}t(1)+7t(2)t(1)-2t(1)+2t(2)^{2}-3t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-13q^{3/2}+20{\sqrt {q}}-{\frac {27}{\sqrt {q}}}+{\frac {30}{q^{3/2}}}-{\frac {30}{q^{5/2}}}+{\frac {25}{q^{7/2}}}-{\frac {18}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+2a^{5}z-3a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z+az^{7}+3az^{5}-z^{5}a^{-1}+6az^{3}-z^{3}a^{-1}+4az+az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+5a^{7}z^{3}-a^{7}z+8a^{6}z^{8}-16a^{6}z^{6}+12a^{6}z^{4}-4a^{6}z^{2}+9a^{5}z^{9}-13a^{5}z^{7}+3a^{5}z^{5}+2a^{5}z^{3}+4a^{4}z^{10}+14a^{4}z^{8}-44a^{4}z^{6}+33a^{4}z^{4}-7a^{4}z^{2}+22a^{3}z^{9}-36a^{3}z^{7}+9a^{3}z^{5}+z^{5}a^{-3}+3a^{3}z^{3}+4a^{2}z^{10}+24a^{2}z^{8}-59a^{2}z^{6}+5z^{6}a^{-2}+33a^{2}z^{4}-2z^{4}a^{-2}-4a^{2}z^{2}+13az^{9}-6az^{7}+13z^{7}a^{-1}-19az^{5}-16z^{5}a^{-1}+13az^{3}+7z^{3}a^{-1}-4az-3za^{-1}+az^{-1}+a^{-1}z^{-1}+18z^{8}-27z^{6}+12z^{4}-2z^{2}-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   9 1   8 2                 11 4     -7 0               16 9       7 -2             16 13         -3 -4           14 14           0 -6         11 16             5 -8       7 14               -7 -10     3 11                 8 -12   1 7                   -6 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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