L11a98

L11a97

L11a99

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Link Presentations

[edit Notes on L11a98's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+7t(2)^{4}+8t(1)t(2)^{3}-9t(2)^{3}-9t(1)t(2)^{2}+8t(2)^{2}+7t(1)t(2)-5t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-15q^{5/2}+19q^{3/2}-22{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}+3a^{3}z^{3}-7az^{3}-3z^{3}a^{-1}+2z^{3}a^{-3}+3a^{3}z-5az-2za^{-1}+2za^{-3}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+5a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-11a^{4}z^{6}+10z^{6}a^{-4}+13a^{4}z^{4}-9z^{4}a^{-4}-5a^{4}z^{2}+5z^{2}a^{-4}-2a^{-4}+4a^{3}z^{9}-11a^{3}z^{7}+15z^{7}a^{-3}+7a^{3}z^{5}-21z^{5}a^{-3}+10z^{3}a^{-3}+2a^{3}z-2za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+5a^{2}z^{8}+14z^{8}a^{-2}-30a^{2}z^{6}-18z^{6}a^{-2}+34a^{2}z^{4}-2z^{4}a^{-2}-12a^{2}z^{2}+10z^{2}a^{-2}+a^{2}-5a^{-2}+12az^{9}+8z^{9}a^{-1}-28az^{7}-z^{7}a^{-1}+14az^{5}-22z^{5}a^{-1}-2az^{3}+13z^{3}a^{-1}+2az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+16z^{8}-47z^{6}+29z^{4}-2z^{2}-3$ (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 12                       1 -1 10                     3   3 8                   7 1   -6 6                 8 3     5 4               11 7       -4 2             11 8         3 0           10 12           2 -2         8 10             -2 -4       4 10               6 -6     3 8                 -5 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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