L11a95

L11a94

L11a96

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Link Presentations

[edit Notes on L11a95's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X18,15,19,16 X16,9,17,10 X10,17,11,18 X22,19,5,20 X20,7,21,8 X8,21,9,22 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 5, -6, 3, -2, 11, -3, 4, -5, 6, -4, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {19}{q^{9/2}}}-{\frac {20}{q^{7/2}}}+{\frac {16}{q^{5/2}}}+q^{3/2}-{\frac {14}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {12}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-7za^{7}-5a^{7}z^{-1}+3z^{5}a^{5}+10z^{3}a^{5}+14za^{5}+8a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-8z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+5a^{10}z^{2}-2a^{10}+4a^{9}z^{7}-3a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{9}z^{-1}+4a^{8}z^{8}+3a^{8}z^{6}-18a^{8}z^{4}+22a^{8}z^{2}-9a^{8}+3a^{7}z^{9}+6a^{7}z^{7}-17a^{7}z^{5}+16a^{7}z^{3}-9a^{7}z+5a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-13a^{6}z^{6}-12a^{6}z^{4}+29a^{6}z^{2}-14a^{6}+7a^{5}z^{9}-27a^{5}z^{5}+33a^{5}z^{3}-21a^{5}z+8a^{5}z^{-1}+a^{4}z^{10}+12a^{4}z^{8}-26a^{4}z^{6}+5a^{4}z^{4}+13a^{4}z^{2}-8a^{4}+4a^{3}z^{9}+2a^{3}z^{7}-24a^{3}z^{5}+25a^{3}z^{3}-14a^{3}z+4a^{3}z^{-1}+6a^{2}z^{8}-12a^{2}z^{6}+3a^{2}z^{4}+2a^{2}z^{2}+4az^{7}-10az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   5 1   -4 -2                 9 3     6 -4               9 7       -2 -6             11 7         4 -8           8 9           1 -10         8 11             -3 -12       4 8               4 -14     2 8                 -6 -16   1 4                   3 -18   2                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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