L11n398

L11n397

L11n399

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Link Presentations

[edit Notes on L11n398's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X15,22,16,19 X7,20,8,21 X19,8,20,9 X13,18,14,5 X11,14,12,15 X17,12,18,13 X21,16,22,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, -3, 9, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(uvw^{3}+uvw^{2}-uvw+uv-uw^{2}+uw+vw^{3}-vw^{2}+w^{4}-w^{3}+w^{2}+w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-5}+4q^{-6}-4q^{-7}+6q^{-8}-5q^{-9}+6q^{-10}-4q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+4a^{12}-5z^{2}a^{10}-5a^{10}z^{-2}-9a^{10}+z^{4}a^{8}+z^{2}a^{8}+2a^{8}z^{-2}+2a^{8}+z^{6}a^{6}+6z^{4}a^{6}+8z^{2}a^{6}+3a^{6}$ (db)
Kauffman polynomial	$z^{7}a^{15}-5z^{5}a^{15}+8z^{3}a^{15}-5za^{15}+a^{15}z^{-1}+2z^{8}a^{14}-9z^{6}a^{14}+12z^{4}a^{14}-8z^{2}a^{14}-a^{14}z^{-2}+4a^{14}+z^{9}a^{13}+z^{7}a^{13}-21z^{5}a^{13}+36z^{3}a^{13}-23za^{13}+5a^{13}z^{-1}+6z^{8}a^{12}-28z^{6}a^{12}+41z^{4}a^{12}-34z^{2}a^{12}-4a^{12}z^{-2}+18a^{12}+z^{9}a^{11}+3z^{7}a^{11}-32z^{5}a^{11}+57z^{3}a^{11}-39za^{11}+9a^{11}z^{-1}+4z^{8}a^{10}-21z^{6}a^{10}+38z^{4}a^{10}-37z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+3z^{7}a^{9}-17z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}-z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}-2a^{8}z^{-2}+5a^{8}-z^{5}a^{7}+z^{3}a^{7}+za^{7}+z^{6}a^{6}-6z^{4}a^{6}+8z^{2}a^{6}-3a^{6}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                       1 1 -9                 2 1     -1 -11               3         3 -13             4 5 1       0 -15           3 1           2 -17         2 4 1           1 -19       4 3               1 -21     1 3                 2 -23   1 3                   -2 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $i=-3$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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