L11a477

L11a476

L11a478

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Link Presentations

[edit Notes on L11a477's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(u-1) (w-1) \left(v^2 w^3-2 v^2 w^2+2 v w^2-2 v w+2 w-1\right)}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q^{-10} -2 q^{-9} +5 q^{-8} -8 q^{-7} +11 q^{-6} -12 q^{-5} +13 q^{-4} -10 q^{-3} +9 q^{-2} -q-5 q^{-1} +3$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^8 z^4+4 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^6-10 a^6 z^4-17 a^6 z^2-2 a^6 z^{-2} -12 a^6+a^4 z^8+6 a^4 z^6+14 a^4 z^4+17 a^4 z^2+a^4 z^{-2} +9 a^4-a^2 z^6-4 a^2 z^4-4 a^2 z^2-a^2$ (db)
Kauffman polynomial	$a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-2 a^{11} z^3+3 a^{10} z^6-2 a^{10} z^4+4 a^9 z^7-5 a^9 z^5+4 a^9 z^3+4 a^8 z^8-6 a^8 z^6+6 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +5 a^8+3 a^7 z^9-3 a^7 z^7-5 a^7 z^5+12 a^7 z^3-10 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+6 a^6 z^8-29 a^6 z^6+43 a^6 z^4-35 a^6 z^2-2 a^6 z^{-2} +15 a^6+6 a^5 z^9-17 a^5 z^7+7 a^5 z^5+11 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+5 a^4 z^8-33 a^4 z^6+51 a^4 z^4-36 a^4 z^2-a^4 z^{-2} +12 a^4+3 a^3 z^9-9 a^3 z^7+a^3 z^5+9 a^3 z^3-3 a^3 z+3 a^2 z^8-13 a^2 z^6+17 a^2 z^4-9 a^2 z^2+2 a^2+a z^7-4 a z^5+4 a z^3-a z$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     2   2 -1                   3 1   -2 -3                 6 2     4 -5               6 5       -1 -7             7 4         3 -9           5 6           1 -11         6 7             -1 -13       2 5               3 -15     3 6                 -3 -17   1 4                   3 -19   1                     -1 -21 1                       1

Integral Khovanov Homology (db, data source)

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

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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