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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 3q-9/2 + 5q-7/2 - 7q-5/2 + 7q-3/2 - 8q-1/2 + 6q1/2 - 5q3/2 + 3q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-14 - q-12 + q-8 + 3q-4 + 3 + q2 + q4 + q6 - q8 - q12

HOMFLY-PT Polynomial:

6a-1z + 11a-1z3 + 6a-1z5 + a-1z7 - az-1 - 13az - 28az3 - 23az5 - 8az7 - az9 + a3z-1 + 6a3z + 11a3z3 + 6a3z5 + a3z7

Kauffman Polynomial:

- 3a-4z2 + 4a-4z4 - a-4z6 + a-3z - 7a-3z3 + 8a-3z5 - 2a-3z7 + a-2z2 - 5a-2z4 + 7a-2z6 - 2a-2z8 - 6a-1z + 16a-1z3 - 16a-1z5 + 9a-1z7 - 2a-1z9 + 6z2 - 6z4 - z6 + 3z8 - z10 + az-1 - 14az + 42az3 - 48az5 + 22az7 - 4az9 - a2 + 6a2z2 - 9a2z4 - a2z6 + 3a2z8 - a2z10 + a3z-1 - 6a3z + 14a3z3 - 18a3z5 + 9a3z7 - 2a3z9 + a4z2 - 6a4z4 + 6a4z6 - 2a4z8 - 2a5z3 + 5a5z5 - 2a5z7 - 3a6z2 + 6a6z4 - 2a6z6 - a7z + 3a7z3 - 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                     1   j = 6                   2 1   j = 4                 3 1     j = 2               3 2       j = 0             5 3         j = -2           3 4           j = -4         4 4             j = -6       2 4               j = -8     1 3                 j = -10   1 2                   j = -12   1                     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, 365]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 365]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[6, 11, 7, 12], X[18, 8, 19, 7], X[22, 18, 11, 17], X[20, 10, 21, 9], > X[8, 20, 9, 19], X[10, 22, 1, 21], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -5, 6, -9, 8, -10}, > {5, -1, 2, -3, 11, -4, 7, -6, 9, -8, 10, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 3 5 7 7 8 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 5 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 9/2 > 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -14 -12 -8 3 2 4 6 8 12 3 + q + q - q + q + -- + q + q + q - q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 365]][a, z]
Out[8]=	3 3 5 a a 6 z 3 11 z 3 3 3 6 z -(-) + -- + --- - 13 a z + 6 a z + ----- - 28 a z + 11 a z + ---- - z z a a a 7 5 3 5 z 7 3 7 9 > 23 a z + 6 a z + -- - 8 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 365]][a, z]
Out[9]=	3 2 2 2 a a z 6 z 3 7 2 3 z z 2 2 -a + - + -- + -- - --- - 14 a z - 6 a z - a z + 6 z - ---- + -- + 6 a z + z z 3 a 4 2 a a a 3 3 4 2 6 2 7 z 16 z 3 3 3 5 3 7 3 > a z - 3 a z - ---- + ----- + 42 a z + 14 a z - 2 a z + 3 a z - 3 a a 4 4 5 5 4 4 z 5 z 2 4 4 4 6 4 8 z 16 z 5 > 6 z + ---- - ---- - 9 a z - 6 a z + 6 a z + ---- - ----- - 48 a z - 4 2 3 a a a a 6 6 3 5 5 5 7 5 6 z 7 z 2 6 4 6 6 6 > 18 a z + 5 a z - a z - z - -- + ---- - a z + 6 a z - 2 a z - 4 2 a a 7 7 8 2 z 9 z 7 3 7 5 7 8 2 z 2 8 > ---- + ---- + 22 a z + 9 a z - 2 a z + 3 z - ---- + 3 a z - 3 a 2 a a 9 4 8 2 z 9 3 9 10 2 10 > 2 a z - ---- - 4 a z - 2 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 2 1 3 2 4 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 4 4 3 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 3 t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 4 2 4 2 q t q t q t 6 4 8 4 10 5 > q t + q t + q t

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