L11a471

L11a470

L11a472

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Link Presentations

[edit Notes on L11a471's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X20,9,21,10 X10,19,11,20 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, 7, -6, 10, -9}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -8, 9, -10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}(t(2)+t(3)-1)(t(3)t(2)-t(2)-t(3))}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}-q^{-5}+9q^{4}+4q^{-4}-14q^{3}-9q^{-3}+21q^{2}+15q^{-2}-22q-20q^{-1}+24$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{6}a^{-2}+2a^{2}z^{4}-2z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-4a^{-2}-z^{6}+3z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-9z^{5}a^{-5}-a^{5}z^{3}+4z^{3}a^{-5}+8z^{8}a^{-4}+4a^{4}z^{6}-22z^{6}a^{-4}-5a^{4}z^{4}+21z^{4}a^{-4}+2a^{4}z^{2}-9z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+7z^{9}a^{-3}+8a^{3}z^{7}-12z^{7}a^{-3}-12a^{3}z^{5}+2z^{5}a^{-3}+7a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z-2za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+9a^{2}z^{8}+15z^{8}a^{-2}-10a^{2}z^{6}-50z^{6}a^{-2}+3a^{2}z^{4}+54z^{4}a^{-2}-3a^{2}z^{2}-27z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+7a^{-2}+6az^{9}+13z^{9}a^{-1}+4az^{7}-20z^{7}a^{-1}-21az^{5}+3z^{5}a^{-1}+14az^{3}+6z^{3}a^{-1}-3az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+16z^{8}-41z^{6}+39z^{4}-23z^{2}-z^{-2}+7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 8 3     -5 5               13 6       7 3             11 10         -1 1           13 11           2 -1         9 13             4 -3       6 11               -5 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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