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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X15,8,16,9 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4}}

Jones Polynomial:

- 2q-9/2 + 4q-7/2 - 8q-5/2 + 9q-3/2 - 10q-1/2 + 10q1/2 - 8q3/2 + 5q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 3q-14 + 2q-10 + 3q-8 - q-6 + 2q-4 - 2q-2 - q4 + 3q6 - q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z - 2a-1z3 - a-1z5 - az3 - az5 - a3z-1 - a3z + a3z3 + a5z-1

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 9a-3z3 + 10a-3z5 - 3a-3z7 - a-2z2 - 2a-2z4 + 7a-2z6 - 3a-2z8 + a-1z - 9a-1z3 + 15a-1z5 - 4a-1z7 - a-1z9 + 2z2 - z4 + 8z6 - 5z8 - 2az + 3az3 + 4az5 - 3az7 - az9 + 2a2z4 - a2z6 - 2a2z8 - a3z-1 + a3z - a3z5 - 2a3z7 + a4 - a4z2 - 2a4z4 - a4z6 - a5z-1 + 3a5z - 3a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               3 1   j = 4             5 2     j = 2           5 3       j = 0         5 5         j = -2       5 6           j = -4     3 4             j = -6   1 5               j = -8 1 3                 j = -10 2

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[11, NonAlternating, 72]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 72]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 16, 8, 17], X[22, 18, 5, 17], > X[18, 12, 19, 11], X[12, 22, 13, 21], X[20, 14, 21, 13], X[14, 20, 15, 19], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, > 8, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 8 9 10 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 10 Sqrt[q] - 8 q + 5 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 2 3 -6 2 2 4 6 8 12 14 q + --- + --- + -- - q + -- - -- - q + 3 q - q + q - q 14 10 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 72]][a, z]
Out[8]=	3 5 3 3 5 a a z 2 z 3 z 2 z 3 3 3 z 5 -(--) + -- + -- - --- - a z + -- - ---- - a z + a z - -- - a z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 72]][a, z]
Out[9]=	3 5 2 2 4 a a z z 3 5 2 2 z z 4 2 a - -- - -- + -- + - - 2 a z + a z + 3 a z + 2 z - ---- - -- - a z - z z 3 a 4 2 a a a 3 3 4 4 9 z 9 z 3 5 3 4 3 z 2 z 2 4 4 4 > ---- - ---- + 3 a z - 3 a z - z + ---- - ---- + 2 a z - 2 a z + 3 a 4 2 a a a 5 5 6 6 7 10 z 15 z 5 3 5 6 z 7 z 2 6 4 6 3 z > ----- + ----- + 4 a z - a z + 8 z - -- + ---- - a z - a z - ---- - 3 a 4 2 3 a a a a 7 8 9 4 z 7 3 7 8 3 z 2 8 z 9 > ---- - 3 a z - 2 a z - 5 z - ---- - 2 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 2 1 3 1 5 3 4 5 5 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 5 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

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