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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X20,11,7,12 X18,13,19,14 X12,19,13,20 X2738 X6,9,1,10

Gauss Code:

{{1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -7, 8, -6}}

Jones Polynomial:

- q-23/2 + 2q-21/2 - 3q-19/2 + 5q-17/2 - 6q-15/2 + 6q-13/2 - 7q-11/2 + 5q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-36 - q-32 + q-30 - q-28 - q-26 + q-24 + 2q-20 + 2q-18 + 2q-16 + 2q-14 - q-12 + q-10 + q-8 - q-6 + q-4

HOMFLY-PT Polynomial:

- a3z - a3z3 - a5z-1 - 2a5z - 2a5z3 + a7z-1 - a7z - 2a7z3 - a9z3 + a11z

Kauffman Polynomial:

a3z - a3z3 + a4z2 - 2a4z4 + a5z-1 - 3a5z + 3a5z3 - 3a5z5 - a6 + a6z2 + 2a6z4 - 3a6z6 + a7z-1 - 3a7z + a7z3 + 4a7z5 - 3a7z7 - 2a8z2 + 4a8z4 + 2a8z6 - 2a8z8 - 3a9z3 + 6a9z5 - a9z9 + 5a10z2 - 15a10z4 + 15a10z6 - 4a10z8 + a11z - 7a11z3 + 4a11z5 + 2a11z7 - a11z9 + 7a12z2 - 15a12z4 + 10a12z6 - 2a12z8 + 2a13z - 7a13z3 + 5a13z5 - a13z7

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                     1 j = -4                   2 1 j = -6                 2     j = -8               3 2     j = -10             4 2       j = -12           2 3         j = -14         4 4           j = -16       2 3             j = -18     1 3               j = -20   1 2                 j = -22   1                   j = -24 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 73]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 73]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 17, 15, 18], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[20, 11, 7, 12], X[18, 13, 19, 14], X[12, 19, 13, 20], > X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -5, 4, -10}, > {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 3 5 6 6 7 5 4 -q + ----- - ----- + ----- - ----- + ----- - ----- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q 2 -(3/2) > ---- - q 5/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -32 -30 -28 -26 -24 2 2 2 2 -12 -10 q - q + q - q - q + q + --- + --- + --- + --- - q + q + 20 18 16 14 q q q q -8 -6 -4 > q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 73]][a, z]
Out[8]=	5 7 a a 3 5 7 11 3 3 5 3 7 3 9 3 -(--) + -- - a z - 2 a z - a z + a z - a z - 2 a z - 2 a z - a z z z
In[9]:=	Kauffman[Link[10, Alternating, 73]][a, z]
Out[9]=	5 7 6 a a 3 5 7 11 13 4 2 6 2 -a + -- + -- + a z - 3 a z - 3 a z + a z + 2 a z + a z + a z - z z 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > 2 a z + 5 a z + 7 a z - a z + 3 a z + a z - 3 a z - 11 3 13 3 4 4 6 4 8 4 10 4 12 4 > 7 a z - 7 a z - 2 a z + 2 a z + 4 a z - 15 a z - 15 a z - 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > 3 a z + 4 a z + 6 a z + 4 a z + 5 a z - 3 a z + 2 a z + 10 6 12 6 7 7 11 7 13 7 8 8 10 8 > 15 a z + 10 a z - 3 a z + 2 a z - a z - 2 a z - 4 a z - 12 8 9 9 11 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 2 1 3 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 24 10 22 9 20 9 20 8 18 8 18 7 16 7 q t q t q t q t q t q t q t 3 4 4 2 3 4 2 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 6 14 6 14 5 12 5 12 4 10 4 10 3 8 3 q t q t q t q t q t q t q t q t 2 2 2 > ----- + ----- + ---- 8 2 6 2 4 q t q t q t

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