L11a291

L11a290

L11a292

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Link Presentations

[edit Notes on L11a291's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X8,9,1,10 X18,12,19,11 X20,13,21,14 X22,16,9,15 X14,7,15,8 X6,22,7,21 X4,18,5,17 X16,6,17,5 X2,19,3,20
Gauss code	{1, -11, 2, -9, 10, -8, 7, -3}, {3, -1, 4, -2, 5, -7, 6, -10, 9, -4, 11, -5, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-2t(1)^{2}t(2)^{2}+2t(1)t(2)^{2}-2t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+5q^{9/2}-10q^{7/2}+15q^{5/2}-21q^{3/2}+22{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}-z^{3}a^{-3}+2za^{-3}+az^{7}+z^{7}a^{-1}+3az^{5}+2z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-4z^{10}a^{-2}-4z^{10}-11az^{9}-19z^{9}a^{-1}-8z^{9}a^{-3}-14a^{2}z^{8}-2z^{8}a^{-2}-5z^{8}a^{-4}-11z^{8}-12a^{3}z^{7}+16az^{7}+54z^{7}a^{-1}+25z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+23a^{2}z^{6}+35z^{6}a^{-2}+15z^{6}a^{-4}+51z^{6}-4a^{5}z^{5}+14a^{3}z^{5}+3az^{5}-37z^{5}a^{-1}-20z^{5}a^{-3}+2z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-10a^{2}z^{4}-34z^{4}a^{-2}-11z^{4}a^{-4}-40z^{4}+2a^{5}z^{3}-4a^{3}z^{3}-9az^{3}-z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-a^{2}z^{2}+5z^{2}a^{-2}+5z^{2}+4az+8za^{-1}+4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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 12                       1 1 10                     4   -4 8                   6 1   5 6                 9 4     -5 4               12 6       6 2             10 9         -1 0           13 12           1 -2         9 12             3 -4       5 11               -6 -6     3 9                 6 -8   1 5                   -4 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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