L10a169

L10a168

L10a170

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Japanese family symbol

Link Presentations

[edit Notes on L10a169's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)^{2}(x-1)^{2}}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+15{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$a^{7}z^{5}+5a^{6}z^{6}-4a^{6}z^{4}+11a^{5}z^{7}-17a^{5}z^{5}+8a^{5}z^{3}-a^{5}z^{-3}+a^{5}z^{-1}+11a^{4}z^{8}-12a^{4}z^{6}+3a^{4}z^{-2}-2a^{4}+4a^{3}z^{9}+19a^{3}z^{7}-48a^{3}z^{5}+z^{5}a^{-3}+24a^{3}z^{3}-3a^{3}z^{-3}+a^{3}z+22a^{2}z^{8}-34a^{2}z^{6}+5z^{6}a^{-2}+8a^{2}z^{4}-4z^{4}a^{-2}+6a^{2}z^{-2}-3a^{2}+4az^{9}+19az^{7}+11z^{7}a^{-1}-48az^{5}-17z^{5}a^{-1}+24az^{3}+8z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}+az+a^{-1}z^{-1}+11z^{8}-12z^{6}+3z^{-2}-2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   4   -4 4                 7 1   6 2               8 4     -4 0             14 7       7 -2           10 12         2 -4         12 10           2 -6       7 14             7 -8     4 8               -4 -10   1 7                 6 -12   4                   -4 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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