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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,15,21,16 X14,19,15,20 X4,18,5,17 X16,6,17,5 X12,21,13,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 8q-9/2 + 13q-7/2 - 17q-5/2 + 19q-3/2 - 19q-1/2 + 15q1/2 - 12q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 2q-14 - 4q-12 + q-10 - 2q-6 + 4q-4 - 2q-2 + 5 + q2 + q4 + 5q6 - q8 + q10 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - 2a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + 3az-1 + 8az + 8az3 + 4az5 + az7 - a3z-1 - 4a3z - 5a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 4a-3z - 9a-3z3 + 9a-3z5 - 3a-3z7 - 2a-2z2 - 2a-2z4 + 8a-2z6 - 4a-2z8 - 2a-1z-1 + 12a-1z - 22a-1z3 + 17a-1z5 - a-1z7 - 3a-1z9 + 3 - 2z2 - 13z4 + 22z6 - 9z8 - z10 - 3az-1 + 11az - 21az3 + 15az5 + 4az7 - 7az9 + 3a2 - 3a2z2 - 12a2z4 + 24a2z6 - 12a2z8 - a2z10 - a3z-1 + 4a3z - 14a3z3 + 19a3z5 - 5a3z7 - 4a3z9 + a4 - 3a4z2 + 2a4z4 + 7a4z6 - 7a4z8 + a5z - 5a5z3 + 11a5z5 - 7a5z7 - 2a6z2 + 6a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 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                   4 1   j = 4                 8 2     j = 2               8 5       j = 0             11 7         j = -2           9 9           j = -4         8 10             j = -6       5 9               j = -8     3 8                 j = -10   1 5                   j = -12   3                     j = -14 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[11, Alternating, 195]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 195]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[18, 13, 19, 14], X[6, 12, 1, 11], X[20, 15, 21, 16], X[14, 19, 15, 20], > X[4, 18, 5, 17], X[16, 6, 17, 5], X[12, 21, 13, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -9, 10, -6}, > {4, -1, 3, -2, 6, -11, 5, -8, 7, -10, 9, -5, 8, -7, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 8 13 17 19 19 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 12 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 4 -10 2 4 2 2 4 6 5 + q - q - q + --- - --- + q - -- + -- - -- + q + q + 5 q - 14 12 6 4 2 q q q q q 8 10 14 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 195]][a, z]
Out[8]=	3 3 3 -2 3 a a 2 z 6 z 3 5 z 6 z 3 --- + --- - -- + --- - --- + 8 a z - 4 a z + a z + -- - ---- + 8 a z - a z z z 3 a 3 a a a 5 3 3 5 3 2 z 5 3 5 7 > 5 a z + a z - ---- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 195]][a, z]
Out[9]=	3 2 4 2 3 a a 4 z 12 z 3 5 2 3 + 3 a + a - --- - --- - -- + --- + ---- + 11 a z + 4 a z + a z - 2 z - a z z z 3 a a 2 2 3 3 2 z 2 z 2 2 4 2 6 2 9 z 22 z 3 > ---- - ---- - 3 a z - 3 a z - 2 a z - ---- - ----- - 21 a z - 4 2 3 a a a a 4 4 3 3 5 3 7 3 4 3 z 2 z 2 4 4 4 > 14 a z - 5 a z + a z - 13 z + ---- - ---- - 12 a z + 2 a z + 4 2 a a 5 5 6 4 9 z 17 z 5 3 5 5 5 7 5 6 > 6 a z + ---- + ----- + 15 a z + 19 a z + 11 a z - a z + 22 z - 3 a a 6 6 7 7 z 8 z 2 6 4 6 6 6 3 z z 7 3 7 > -- + ---- + 24 a z + 7 a z - 4 a z - ---- - -- + 4 a z - 5 a z - 4 2 3 a a a a 8 9 5 7 8 4 z 2 8 4 8 3 z 9 3 9 > 7 a z - 9 z - ---- - 12 a z - 7 a z - ---- - 7 a z - 4 a z - 2 a a 10 2 10 > z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 8 5 9 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 10 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 8 q t + 5 q t + 8 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 2 q t + q t

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