L11a237

L11a236

L11a238

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Link Presentations

[edit Notes on L11a237's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X14,12,15,11 X22,15,7,16 X16,9,17,10 X10,21,11,22 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -5, 8, -7, 11, -8, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-5u^{2}v+u^{2}-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u+v^{4}-5v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$-11q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {3}{q^{7/2}}}-23q^{5/2}+{\frac {8}{q^{5/2}}}+25q^{3/2}-{\frac {15}{q^{3/2}}}-q^{13/2}+5q^{11/2}-25{\sqrt {q}}+{\frac {20}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}-2a^{3}z+6az-4za^{-1}+za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-13z^{9}a^{-1}-8z^{9}a^{-3}-5a^{2}z^{8}-21z^{8}a^{-2}-13z^{8}a^{-4}-13z^{8}-3a^{3}z^{7}+2az^{7}+12z^{7}a^{-1}-4z^{7}a^{-3}-11z^{7}a^{-5}-a^{4}z^{6}+8a^{2}z^{6}+46z^{6}a^{-2}+17z^{6}a^{-4}-5z^{6}a^{-6}+33z^{6}+7a^{3}z^{5}+12az^{5}+16z^{5}a^{-1}+27z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-a^{2}z^{4}-25z^{4}a^{-2}-4z^{4}a^{-4}+4z^{4}a^{-6}-21z^{4}-6a^{3}z^{3}-16az^{3}-21z^{3}a^{-1}-16z^{3}a^{-3}-5z^{3}a^{-5}-3a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}+3a^{3}z+10az+9za^{-1}+2za^{-3}+a^{4}+3a^{2}+3-a^{3}z^{-1}-3az^{-1}-2a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   7 1   6 8                 10 4     -6 6               13 7       6 4             13 11         -2 2           12 12           0 0         9 14             5 -2       6 11               -5 -4     2 9                 7 -6   1 6                   -5 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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