L11a494

L11a493

L11a495

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Link Presentations

[edit Notes on L11a494's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)^{3}\left(w^{2}+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-12+19q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-3a^{6}z^{2}-a^{6}z^{-2}-3a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+12a^{4}z^{2}+4a^{4}z^{-2}+10a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-12a^{2}z^{2}-5a^{2}z^{-2}-11a^{2}+z^{6}+3z^{4}+3z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-4a^{8}z^{4}+a^{8}z^{2}+6a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+7a^{6}z^{8}-10a^{6}z^{6}+8a^{6}z^{4}-9a^{6}z^{2}-a^{6}z^{-2}+5a^{6}+5a^{5}z^{9}+a^{5}z^{7}-20a^{5}z^{5}+32a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+2a^{4}z^{10}+11a^{4}z^{8}-38a^{4}z^{6}+52a^{4}z^{4}-38a^{4}z^{2}-4a^{4}z^{-2}+16a^{4}+12a^{3}z^{9}-25a^{3}z^{7}+8a^{3}z^{5}+25a^{3}z^{3}-26a^{3}z+9a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}-50a^{2}z^{6}+z^{6}a^{-2}+65a^{2}z^{4}-2z^{4}a^{-2}-40a^{2}z^{2}-5a^{2}z^{-2}+17a^{2}+7az^{9}-16az^{7}+4z^{7}a^{-1}+8az^{5}-9z^{5}a^{-1}+5az^{3}+2z^{3}a^{-1}-10az+5az^{-1}+8z^{8}-24z^{6}+23z^{4}-12z^{2}-2z^{-2}+7$ (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 7                       1 1 5                     3   -3 3                   6 1   5 1                 6 3     -3 -1               13 6       7 -3             11 10         -1 -5           10 9           1 -7         7 11             4 -9       6 10               -4 -11     2 7                 5 -13   1 6                   -5 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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