L11a363

L11a362

L11a364

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Link Presentations

[edit Notes on L11a363's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,4,15,3 X22,14,11,13 X2,11,3,12 X4,22,5,21 X20,10,21,9 X16,6,17,5 X8,18,9,17 X18,8,19,7 X6,20,7,19 X10,16,1,15
Gauss code	{1, -4, 2, -5, 7, -10, 9, -8, 6, -11}, {4, -1, 3, -2, 11, -7, 8, -9, 10, -6, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^4 v^2-u^4 v+2 u^3 v^3-6 u^3 v^2+4 u^3 v-u^3+u^2 v^4-6 u^2 v^3+9 u^2 v^2-6 u^2 v+u^2-u v^4+4 u v^3-6 u v^2+2 u v-v^3+v^2}{u^2 v^2}$ (db)
Jones polynomial	$q^{21/2}-3 q^{19/2}+6 q^{17/2}-11 q^{15/2}+14 q^{13/2}-17 q^{11/2}+17 q^{9/2}-15 q^{7/2}+11 q^{5/2}-7 q^{3/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^5 a^{-3} -2 z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} -3 z^3 a^{-5} -z^3 a^{-7} +z^3 a^{-9} +z a^{-1} +2 z a^{-3} -z a^{-7} +z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1}$ (db)
Kauffman polynomial	$-z^{10} a^{-6} -z^{10} a^{-8} -3 z^9 a^{-5} -6 z^9 a^{-7} -3 z^9 a^{-9} -5 z^8 a^{-4} -7 z^8 a^{-6} -6 z^8 a^{-8} -4 z^8 a^{-10} -5 z^7 a^{-3} -3 z^7 a^{-5} +7 z^7 a^{-7} +2 z^7 a^{-9} -3 z^7 a^{-11} -3 z^6 a^{-2} +5 z^6 a^{-4} +11 z^6 a^{-6} +13 z^6 a^{-8} +9 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +8 z^5 a^{-3} +9 z^5 a^{-5} -3 z^5 a^{-7} +6 z^5 a^{-9} +9 z^5 a^{-11} +5 z^4 a^{-2} -z^4 a^{-4} -4 z^4 a^{-6} -5 z^4 a^{-8} -4 z^4 a^{-10} +3 z^4 a^{-12} +2 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} +6 z^3 a^{-7} -6 z^3 a^{-9} -8 z^3 a^{-11} -2 z^2 a^{-2} +z^2 a^{-6} -z^2 a^{-10} -2 z^2 a^{-12} -z a^{-1} +2 z a^{-3} -3 z a^{-5} -5 z a^{-7} +3 z a^{-9} +2 z a^{-11} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   4 1   -3 16                 7 2     5 14               8 5       -3 12             9 6         3 10           8 8           0 8         7 9             -2 6       4 8               4 4     3 7                 -4 2   1 5                   4 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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