L11a435

L11a434

L11a436

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Link Presentations

[edit Notes on L11a435's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)\left(-uvw^{3}+4uvw^{2}-4uvw+2uv+uw^{3}-2uw^{2}+2uw+2v^{2}w^{2}-2v^{2}w+v^{2}+2vw^{3}-4vw^{2}+4vw-v\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+13q^{5}-18q^{4}+21q^{3}-20q^{2}+19q-13+9q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-8}+3z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}-3z^{4}a^{-4}-2z^{2}a^{-4}+4a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-9a^{-2}-2z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+4z^{9}a^{-1}+8z^{9}a^{-3}+4z^{9}a^{-5}+16z^{8}a^{-2}+16z^{8}a^{-4}+6z^{8}a^{-6}+6z^{8}+3az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}+7z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-46z^{6}a^{-2}-38z^{6}a^{-4}-5z^{6}a^{-6}+3z^{6}a^{-8}-15z^{6}-6az^{5}-32z^{5}a^{-1}-48z^{5}a^{-3}-29z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+52z^{4}a^{-2}+35z^{4}a^{-4}-z^{4}a^{-6}-5z^{4}a^{-8}+18z^{4}+3az^{3}+39z^{3}a^{-1}+68z^{3}a^{-3}+37z^{3}a^{-5}+3z^{3}a^{-7}-2z^{3}a^{-9}+3a^{2}z^{2}-44z^{2}a^{-2}-22z^{2}a^{-4}+4z^{2}a^{-6}+3z^{2}a^{-8}-18z^{2}-23za^{-1}-41za^{-3}-21za^{-5}-2za^{-7}+za^{-9}-a^{2}+23a^{-2}+14a^{-4}-a^{-8}+10+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   5 1   -4 11                 8 2     6 9               11 6       -5 7             10 7         3 5           10 11           1 3         9 10             -1 1       6 12               6 -1     3 7                 -4 -3     6                   6 -5 1 3                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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